Integrated process-structure-property modeling frameworks and methods for design optimization and/or performance prediction of material systems and applications of same

ABSTRACT

Integrated process-structure-property modeling framework and method for design optimization and/or performance prediction of a material system are provided. The Integrated process-structure-property modeling framework includes a powder spreading model using a discrete element method to generate a powder bed; a thermal-fluid flow model of the powder melting process to predict voids and temperature profile; a cellular automaton model to simulate grain growth based on the temperature profile; and a reduced-order micromechanics model to predict mechanical properties and fatigue resistance of resultant structures by resolving the voids and grains.

CROSS-REFERENCE TO RELATED PATENT APPLICATION

This application claims priority to and the benefit of, pursuant to 35U.S.C. § 119(e), of U.S. provisional patent application Ser. No.62/731,381, filed Sep. 14, 2018, entitled “MULTISCALE MODELING PLATFORMAND APPLICATIONS OF SAME”, by Wing Kam Liu, Jiaying Gao, Cheng Yu andOrion L. Kafka, which is incorporated herein by reference in itsentirety.

This application is related to a co-pending PCT patent application,entitled “DATA-DRIVEN REPRESENTATION AND CLUSTERING DISCRETIZATIONMETHOD AND SYSTEM FOR DESIGN OPTIMIZATION AND/OR PERFORMANCE PREDICTIONOF MATERIAL SYSTEMS AND APPLICATIONS OF SAME”, by Wing Kam Liu, JiayingGao, Cheng Yu, and Orion L. Kafka, with Attorney Docket No.0116936.152WO2, filed on the same day that this application is filed,and with the same assignee as that of this application, which isincorporated herein by reference in its entirety.

FIELD OF THE INVENTION

The invention relates generally to materials, and more particularly, tointegrated process-structure-property modeling frameworks and methodsfor design optimization and/or performance prediction of materialsystems and applications of the same.

BACKGROUND OF THE INVENTION

The background description provided herein is for the purpose ofgenerally presenting the context of the invention. The subject matterdiscussed in the background of the invention section should not beassumed to be prior art merely as a result of its mention in thebackground of the invention section. Similarly, a problem mentioned inthe background of the invention section or associated with the subjectmatter of the background of the invention section should not be assumedto have been previously recognized in the prior art. The subject matterin the background of the invention section merely represents differentapproaches, which in and of themselves may also be inventions. Work ofthe presently named inventors, to the extent it is described in thebackground of the invention section, as well as aspects of thedescription that may not otherwise qualify as prior art at the time offiling, are neither expressly nor impliedly admitted as prior artagainst the invention.

Metal-based AM is widely considered a promising technology, with manypotential applications due to the flexibility of the process. Thisflexibility is achieved in part because the process is highly localized;however, these processes lack reproducibility and reliability: thequality of the parts produced are sensitive to build conditions. Toovercome the challenges this poses in the production of functional,load-bearing or otherwise practically useful components, significanteffort has been expended. Computational modeling is an appealingapproach to understand the complex relationships between the processingconditions and part quality. Recent researches have called for theintegration of process-structure and structure-property-performancetools for predictive computational modeling to further enable AM. Byproviding predictive models that mimic the stages of production and useof a part, these tools could allow for higher confidence in AM parts andan integration of part and process design to maximize the potential ofAM.

In addition, multiscale simulation methods present significantadvantages for computational mechanics due to their ability to analyzemacroscopic structural performance while considering the effects ofmicroscopic heterogeneities. While great strides have been made, somecore challenges still face multiscale methods, including accuratemethods for homogenizing the microstructural representative volumeelement (RVE) undergoing strain localization, computation of strainlocalization with variable-sized microstructural features, and theability to conduct efficient concurrent simulations.

Therefore, a heretofore unaddressed need exists in the art to addressthe aforementioned deficiencies and inadequacies.

SUMMARY OF THE INVENTION

In one aspect, the invention relates to an integratedprocess-structure-property modeling framework for design optimizationand/or performance prediction of a material system. In one embodiment,the framework includes a database; and a plurality of models coupledwith the database, comprising process models and mechanical responsemodels, each model being coupled to the next by passing outputinformation to a subsequent input processor. The process models comprisea powder spreading model, a powder melting model and a grain growthmodel. The mechanical response models comprise a self-consistentclustering analysis (SCA) model, a finite element method (FEM) modeland/or a fatigue prediction model.

The powder spreading model and the powder melting model operablyexchange geometry information of a powder bed using a stereolithography(STL)-format surface mesh representation. In one embodiment, the powderspreading model is a discrete element method (DEM) powder spreadingmodel.

The powder melting model operably predicts a temperature profile in eachvolume of interest (VOI) with a powder melting simulation. Thetemperature profile is written to the database in a flat file format,and a void space is set to an ambient temperature to distinguish saidvoid space from a dense material. In one embodiment, the powder meltingmodel is a computational fluid dynamics (CFD) powder melting model.

The grain growth model operably accesses the temperature profile andvoid information from the database, and predicts grain information basedon the temperature profile and void information. The grain informationis written to the database along with the voids. The dense material andthe voids are identified based on the temperature profile, and the voidsand free surfaces are thereby included in the grain growth model. In oneembodiment, the grain growth model is a cellular automaton (CA) graingrowth model. In one embodiment, the grain information comprises phaseand orientation.

The SCA and/or FEM models are used, based on the grain and voidinformation provided at each point via the database, to predictstress-strain responses. The stress-strain responses are written to thedatabase. In one embodiment, the SCA model is an SCA crystal plasticitymodel.

The fatigue prediction model operably accesses the stress-strainresponses at each point via the database, and computes a non-localpotency estimate based on the stress-strain responses. The fatigue lifeof the highest potency location is computed to provide a scalar materialresponse metric.

In one embodiment, in operation, the powder spreading model generates aparticle packing configuration within one powder layer with a particlespreading simulation, outputs the particle packing configuration as afirst STL file, and feeds the particle packing configuration into thepowder melting model, thereby reproducing powder spreading and melting;and the powder melting model predicts a solidified shape with the powdermelting simulation, outputs the solidified shape as a second STL file,and feeds the solidified shape into the powder spreading model to applya new powder layer, thereby reproducing powder spreading.

In one embodiment, the manufacturing process of multiple layers andtracks comprises repeating said two simulations and the data exchangebetween the powder spreading model and the powder melting model.

In one embodiment, for the mechanical response models, a sub-domain ofparticular interest of the full VOI for grain growth is selected as aninput, the grain and void information provided at each point via thedatabase is assigned as the center point of each voxel in a regular,uniform, cuboid mesh, and said mesh is used for the mechanical responsemodels.

In one embodiment, in operation, the powder melting model feeds thetemperature profiles and void information to the grain growth model viathe database, and the grain growth model feeds the grain and voidinformation to the SCA model via the database.

In one embodiment, a material region is discretized by a set of cubiccells, and the temperature history of each cell is determined from athermal-CFD simulation using a linear interpolation.

In another aspect, the invention relates to a method of integratedprocess-structure-property modeling for design optimization and/orperformance prediction of a material system. In one embodiment, themethod includes exchanging geometry information of a powder bed betweena powder spreading model and a powder melting model using an STL-formatsurface mesh representation; predicting a temperature profile in eachVOI by the powder melting model with a powder melting simulation;writing the temperature profile to the database in a flat file format;and setting a void space to an ambient temperature to distinguish saidvoid space from a dense material; accessing the temperature profile andvoid information from the database and predicting grain information by agrain growth model based on the temperature profile and voidinformation; writing the grain information to the database along withthe voids; identifying the dense material and the voids on thetemperature profile, thereby including the voids and free surfaces inthe grain growth model; predicting stress-strain responses by the SCAand/or FEM models based on the grain and void information provided ateach point via the database; and writing the stress-strain responses tothe database; and accessing the stress-strain responses at each pointvia the database and computing a non-local potency estimate by a fatigueprediction model based on the stress-strain responses, wherein thefatigue life of the highest potency location is computed to provide ascalar material response metric.

In one embodiment, said exchanging the geometry information comprises,by the powder spreading model, generating a particle packingconfiguration within one powder layer with a particle spreadingsimulation, outputing the particle packing configuration as a first STLfile, and feeding the particle packing configuration into the powdermelting model, thereby reproducing powder spreading and melting; and, bythe powder melting model, predicting a solidified shape with a powdermelting simulation, outputing the solidified shape as a second STL file,and feeding the solidified shape into the powder spreading model toapply a new powder layer, thereby reproducing powder spreading.

In one embodiment, said exchanging the geometry information furthercomprises repeating said two simulations and the data exchange betweenthe powder spreading model and the powder melting model, for formingmultiple layers and tracks.

In one embodiment, the particle packing configuration is generated usingthe DEM.

In one embodiment, the powder melting model is a CFD powder meltingmodel.

In one embodiment, the grain growth model is a CA grain growth model.

In one embodiment, the SCA model is an SCA crystal plasticity model.

In yet another aspect, the invention relates to a non-transitorytangible computer-readable medium storing instructions which, whenexecuted by one or more processors, cause a system to perform the abovemethods of integrated process-structure-property modeling for designoptimization and/or performance prediction of a material system.

In a further aspect, the invention relates to a computational system fordesign optimization and/or performance prediction of a material system,which includes one or more computing devices comprising one or moreprocessors; and a non-transitory tangible computer-readable mediumstoring instructions which, when executed by the one or more processors,cause the one or more computing devices to perform the above methods ofintegrated process-structure-property modeling for design optimizationand/or performance prediction of a material system.

In one aspect, the invention relates to an integratedprocess-structure-property modeling framework for design optimizationand/or performance prediction of a material system, comprising a powderspreading model using the DEM to generate a powder bed; a thermal-fluidflow model of the powder melting process to predict voids andtemperature profile; a CA model to simulate grain growth based on thetemperature profile; and a reduced-order micromechanics model to predictmechanical properties and fatigue resistance of resultant structures byresolving the voids and grains.

In one embodiment, in the CA, a material region is discretized by a setof cubic cells, and the temperature history of each cell is determinedfrom a thermal-CFD simulation using a linear interpolation.

In one embodiment, each model incorporates basic material informationand data provided directly from previous models in the framework topredict the mechanical response and estimated fatigue life of criticalmicrostructures given machine-relevant processing conditions.

In another aspect, the invention relates to method for implementation ofmultiresolution continuum theory (MCT). In one embodiment, the methodincludes decomposing deformation across different microstructural lengthscales, by an MCT model; wherein the macroscale deformation is describedby conventional degrees of freedom and extra degrees of freedom areintroduced to describe the deformation at a microscale; employing amodified homogenization-based Gurson (GTN) model at the macroscale toresult in equations for a void volume fraction, an effective plasticstrain rate, and a matrix flow stress, wherein an additive decompositionof a rate of deformation tensor into elastic and plastic parts allowsfor description of an objective rate of Cauchy stress; solving theresulting equations at each scale using an iterative Newton-Raphsonscheme to update the void volume fraction, the effective plastic strainrate, and the matrix flow stress based on a new deformation tensor,wherein the homogenized void growth behavior is taken to represent theresponse of the primary population of voids, including those initiatedat inclusions, as the material is deformed; and constructing a plasticpotential using microstress and microstress couple at the microscale,wherein a deviatoric strain rate is defined in terms of a relativedeformation between the microscale and the macroscale, and solvingequations of the plastic potential to obtain stress-strain profiles. Inone embodiment, the material structure and the deformation field areresolved at each scale;

the resulting internal power is a multi-field expression withcontributions from the average deformation at each scale; and thedeformation behavior at each scale is found by testing themicromechanical response.

In one embodiment, the MCT model is implemented with a user elementsubroutine including a FEA solver; user materials subroutines for microand macro domains; an auxiliary function subroutine containing functionsused by the user materials subroutines needed for the user elementsubroutine, wherein the auxiliary function subroutine is callable by theuser material subroutines to finish the update of the stress and strain;a force subroutine for calculating the internal force by calling theuser materials subroutines and the auxiliary function subroutine; aparameter subroutine providing input parameters for the user elementsubroutine; and a mass subroutine for computing a mass matrix for theuser element subroutine, wherein the force subroutine, parametersubroutine and the mass subroutines are called by the user elementsubroutine for solving the equations at each scale.

In one embodiment, the auxiliary function subroutine comprises Gaussintegration, computation of inertia and subroutines to calculate thestrain and strain rate.

In one embodiment, the input parameters include number of micro scales,number of Gauss points and number of history variables.

In one embodiment, the method is applicable to prediction of formationand propagation of a shear band in a material system.

In one embodiment, the method is applicable to simulation of high speedmetal cutting (HSMC) in a material system.

In another aspect, the invention relates to a non-transitory tangiblecomputer-readable medium storing instructions which, when executed byone or more processors, cause a system to perform the above disclosedmethods for implementation of the MCT.

In yet another aspect, the invention relates to a computational systemfor design optimization and/or performance prediction of a materialsystem, comprising one or more computing devices comprising one or moreprocessors; and a non-transitory tangible computer-readable mediumstoring instructions which, when executed by the one or more processors,cause the one or more computing devices to perform above disclosedmethods for implementation of the MCT.

In a further aspect, the invention relates to a method of data-drivenmechanistic modeling of a system for predicting high cycle fatigue (HCF)crack incubation life. In one embodiment, the method comprisesgenerating a representation of the system with matrix, inclusion andvoid, wherein the representation comprises microstructure volumeelements (MVE) that are of building blocks of the system; obtainingstrain contours in the matrix from a linear elastic analysis over apredefined set of boundary conditions; generating clusters from a strainsolution by assigning voxels with similar strain concentration tensorwithin the representation of the material system to one of the clusterswith a unique ID, using k-means clustering; computing a plastic shearstrain field of the material system by solving the Lippmann-Schwingerequation using the generated clusters with a crystal plasticity (CP)model; and predicting the fatigue crack incubation life of the systemusing a fatigue indicating parameter (FIP) based on the plastic shearstrain field.

In one aspect, the invention relates to a non-transitory tangiblecomputer-readable medium storing instructions which, when executed byone or more processors, cause a system to perform the above discloseddata-driven mechanistic modeling methods for predicting high cyclefatigue (HCF) crack incubation life.

In another aspect, the invention relates to computational system fordesign optimization and/or performance prediction of a material system,comprising one or more computing devices comprising one or moreprocessors; and a non-transitory tangible computer-readable mediumstoring instructions which, when executed by the one or more processors,cause the one or more computing devices to perform the above discloseddata-driven mechanistic modeling methods for predicting high cyclefatigue (HCF) crack incubation life.

These and other aspects of the invention will become apparent from thefollowing description of the preferred embodiment taken in conjunctionwith the following drawings, although variations and modificationstherein may be affected without departing from the spirit and scope ofthe novel concepts of the invention.

BRIEF DESCRIPTION OF THE DRAWINGS

The patent or application file contains at least one drawing executed incolor. Copies of this patent or patent application publication withcolor drawing(s) will be provided by the Patent and Trademark Officeupon request and payment of the necessary fee.

The following drawings form part of the present specification and areincluded to further demonstrate certain aspects of the invention. Theinvention may be better understood by reference to one or more of thesedrawings in combination with the detailed description of specificembodiments presented herein. The drawings described below are forillustration purposes only. The drawings are not intended to limit thescope of the present teachings in any way.

FIG. 1A shows a schematic diagram of the module-hub framework conceptuallayout according to embodiments of the invention, with images of ourcurrent implementation demonstrating the modules: a SEBM process withmicrostructure-based fatigue life assessment. The data used by theframework at each point in space is described by spatial coordinates andany relevant field data for the current working modules, as showngraphically by a schematic of a representative voxel. The cellularautomata model is shaded gray to indicate that results both with andwithout it are given, to demonstrate the flexibility of the framework.

FIG. 1B shows a schematic diagram of data flow within the one-waycoupled framework according to embodiments of the invention. Arrows withthe same color show the data flow between models.

FIG. 2 shows a schematic of the flat-file database used to store recorddata for the framework according to embodiments of the invention. Thefields are abbreviated to more clearly show the structure of thedatabase.

FIG. 3 shows a schematic of the Z-pattern electron beam scan strategyaccording to embodiments of the invention.

FIG. 4 shows powder bed configurations generated by powder spreadingsimulations according to embodiments of the invention. Panel (a), thefirst powder layer with a thickness of 0.05 mm spread on the substrate.Panel (b), the second powder layer with a thickness of 0.05 mm spread onthe solidified surface of the first layer. The figures are colored bythe height from the substrate surface. In panels (a) and (b), the uppersub-figures show views from above, and the lower ones show views frombelow (the −z direction), with the non-powder material hidden.

FIG. 5 shows an example temperature history taken from a single point ofthe thermal-CFD simulation according to embodiments of the invention.The inset curve highlights the kink caused by latent heat.

FIG. 6 shows voids observed in the CFD simulation of multiple-layermultiple-track manufacturing process according to embodiments of theinvention. This is a 3D view of the free surfaces after manufacturing,colored by the height from the substrate surface. The upper image showsa view from above, and the lower image shows a view from below, with thesubstrate material hidden. The inset image highlights a void predictedduring the build.

FIG. 7 shows a molten pool profile colored by temperature at t=0.157 ms(panel (a)) and evolution of the subsequent solidificationmicrostructure represented by three snapshots at relative time: t=0.234ms (panel (b)), t=0.438 ms (panel (c)), and t=0.931 ms (panel (d)),according to embodiments of the invention. Grain coloring is selected tomaximize contrast between nearby grains.

FIG. 8 shows a longitudinal cross section view of molten poolmicrostructure to show angled grain growth following the direction ofmotion of the heat source according to embodiments of the invention.

FIG. 9 shows a transverse cross section view of molten poolmicrostructure to show the radial distribution of grain growth accordingto embodiments of the invention.

FIG. 10 shows embodiments of the invention: panel (a), a 2D section ofthe CA result with the relative locations of four cubic VOIs atdifferent locations through the height of a two-layer build (color codedgreen, purple, and black to upper, middle, and lower) and one near theintersection of two tracks of the same build (red; for FIP with/withoutvoid), where the white dashed lines show approximate melt regionboundaries for each layer, panel (b), these VOIs in 3D with thesurrounding material removed, panel (c), the single crystal voxel meshused for FIP prediction, panel (d), clustering used for the singlecrystal, with a void (white) case, panel (e), the clusters used for thepolycrystal case, and panel (f), the clusters used for the polycrystalwith void case.

FIG. 11 shows stress-strain curves predicted by SCA with CP at thestrain rate of 10⁻⁴s⁻¹, according to embodiments of the invention. Panel(a) is for the upper VOI showing the anisotropy of overall response, forall VOIs. The response for each VOI is given for x-direction loading inpanel (b), y-direction loading in panel (c), and z-direction loading inpanel (d).

FIG. 12 shows experimental strain life data (points) for HCF, and fourpredictions of fatigue life according to embodiments of the invention:(1) a single crystal without any defects (baseline data), (2) a singlecrystal with a void predicted with the thermal-CFD model, (3) the fullpolycrystalline fatigue VOI shown in FIG. 10, and (4) this same growthpattern, including a void.

FIG. 13 shows a framework for the three-scale homogenization accordingto embodiments of the invention.

FIG. 14 shows relationships of subroutines used in MCT implementationaccording to embodiments of the invention.

FIG. 15 shows a dog-bone model used to test our implementation accordingto embodiments of the invention. The bottom is fixed and the top isgiven a displacement.

FIG. 16 shows comparison of a user element and a standard elementaccording to embodiments of the invention.

FIG. 17 shows comparison of the results with and without microscaleaccording to embodiments of the invention.

FIG. 18 shows influence of microscale parameters for a purely elasticbehavior used for the microscale (panel a) Young's modulus and (panel b)length scale according to embodiments of the invention.

FIG. 19 shows influence of microscale parameters when plastic behaviorwith hardening is used for the microscale (panel a) Young's modulus and(panel b) length scale, according to embodiments of the invention.

FIG. 20 shows influence of microscale parameters for perfectly plasticbehavior in the microscale (panel a) Young's modulus and (panel b)length scale, according to embodiments of the invention.

FIG. 21 shows influence of the microscale parameters on a thermalsoftening macroscale response for an elastic microscale material (panela) Young's modulus and (panel b) length scale, according to embodimentsof the invention.

FIG. 22 shows an effect of the microscale parameters on macroscalethermal softening when plastic behavior without hardening is used inmicroscale (panel a) Young's modulus and (panel b) length scale,according to embodiments of the invention.

FIG. 23 shows comparison of efficiency and speedup under strong scalingfor standard elements, MCT with no microscales, and MCT with onemicroscale, according to embodiments of the invention. The user elementreduces the relative impact of communication and thus allows for a lowerelement count/processor count ratio.

FIG. 24 shows a finite element mesh for a simple shear geometryaccording to embodiments of the invention.

FIG. 25 shows shear band formation with conventional continuum FEM,fringes show equivalent plastic strain (panel a) coarse mesh and (panelb) finer mesh, according to embodiments of the invention.

FIG. 26 shows simulation results under different length scale l(E⁽¹⁾=0.01E), fringes show equivalent plastic strain a) l=0.1 mm b)l=0.04 mm c) l=0.02 mm, according to embodiments of the invention.

FIG. 27 shows comparison of shear band width for different mesh size,fringes show equivalent plastic strain (panel a) mesh size 0.01 and(panel b) mesh size 0.005, according to embodiments of the invention.

FIG. 28 shows MCT simulations with different numbers of microscalesaccording to embodiments of the invention: (panel a), a MCT simulationwith two microscales (E⁽¹⁾=0.01E, l₁=0.1 mm, E⁽²⁾=0.1E⁽¹⁾, l₂=0.05 mm),(panel b), a MCT simulation with one microscale (E⁽¹⁾=0.01E, l₁=0.1 mm),and (panel c), equivalent plastic strain distributions across the shearband.

FIG. 29 shows geometry model of metal cutting according to embodimentsof the invention.

FIG. 30 shows (panel a) contours of J2 stress at 0.2 ms, noting thecharacteristic saw-tooth chip with stress localized about the shearbands, and (panel b) serrated chip formation, according to embodimentsof the invention.

FIG. 31 shows macrostress (left) and microstress (left) XY componentscontours at 0.22 ms according to embodiments of the invention.

FIG. 32 shows magitudes of effective plastic strain and dotsrepresenting the current cutting force in each pane for four differentstages of the cut, according to embodiments of the invention: (panel a),at the formation of the first shear band (0.08 ms), (panel b), thebeginning of formation of the second shear band (0.12 ms), and (panelc), second shear band fully formed (0.14 ms), and the beginning offormation of the third shear band (0.16 ms). Clearly, formation of shearband is associated with force drop.

FIG. 33 shows a schematic showing the essential components of the SCAmethod according to embodiments of the invention. The figure shows anexample RVE from the analysis conducted for the parametric study, withhalf of the matrix phase shown and the inclusion and void phases hidden.

FIG. 34 shows 3D rendering of the RVE microstructure, with the matrixmaterial hidden and a 2D slice showing the parameters void width d andinclusion diameter D, according to embodiments of the invention.

FIG. 35 shows a response surface of fatigue crack incubation life tovoid width and inclusion diameter according to embodiments of theinvention. This shows that the estimated number of crack incubationcycles decreases with increasing inclusion diameter but increases withincreasing void width between inclusions.

FIG. 36 shows illustration of the RVE microstructural stress effects ofinclusions in the material microstructure (left) during localization ofa macroscale model (right).

FIG. 37 shows a flowchart summarizing the self-consistent clusteringanalysis (SCA) for model reduction according to embodiments of theinvention. Clusters in the inclusion phase are not shown.

FIG. 38 shows illustration of the micro-damage algorithm on themacroscopic stress-strain curve according to embodiments of theinvention.

FIG. 39 shows K-means clustering results of the cross-section of atwo-dimensional RVE with circular inclusions according to embodiments ofthe invention. The high-fidelity RVE is discretized by a 1200×1200 mesh.Each cluster contains all the separate sub-domains with the same color.

FIG. 40 shows illustration of the microscale RVE of a two-phaseheterogeneous material used for the analysis according to embodiments ofthe invention: matrix (phase 1) with randomly distributed circularinclusions (phase 2) embedded. The DNS with 1200×1200 FE mesh has 1.44million 4-node linear plane strain element with reduced integration.

FIG. 41 shows the DNS results for the micro-damage algorithm of adamaged RVE with hard inclusions (left) and soft inclusions (right)under uniaxial loading according to embodiments of the invention. Threedifferent RVE sizes are investigated: L/2, L, and 2L.

FIG. 42 shows the stress-strain curves of the undamaged RVE with hardinclusions (left) and soft inclusions (right) according to embodimentsof the invention. Three macro-strain loading conditions are considered:uniaxial loading (blue), biaxial loading (black) and shear loading(red). The solid lines represent the DNS results for comparison.

FIG. 43 shows the stress-strain curves of the damaged RVE with hardinclusions for uniaxial tension, biaxial tension, and shear loadingaccording to embodiments of the invention.

FIG. 44 shows the stress-strain curves of the damaged RVE with softinclusions for uniaxial tension, biaxial tension, and shear loadingaccording to embodiments of the invention.

FIG. 45 shows the material toughness calculations for RVEs with hardinclusions (left) and soft inclusions (right) according to embodimentsof the invention.

FIG. 46 shows the stress-strain curves of the damaged RVE with hardinclusions (left) and soft inclusions (right) for uniaxial tensionloading according to embodiments of the invention. Damage parameters arecalibrated using energy regularization.

FIG. 47 shows RVE damage fields in the matrix phase at {ε_(M)}₁₁=0.04predicted by DNS and SCA following energy regularization for hardinclusions and soft inclusions according to embodiments of theinvention. No damage is considered in the inclusion phase.

FIG. 48 shows the material toughness calculations following energyregularization for hard inclusions (left) and soft inclusions (right)according to embodiments of the invention. The results without energyregularization are plotted for reference, and are denoted as (NR).

FIG. 49 shows geometry of the double-notched specimen in the 2Dplane-strain condition according to embodiments of the invention.

FIG. 50 shows load-displacement curves with different mesh size l_(e)for hard inclusions according to embodiments of the invention. Thenon-local length parameter l₀ is 2 mm.

FIG. 51 shows crack patterns with different mesh size l_(e) for hardinclusions according to embodiments of the invention. The non-locallength parameter is l₀=2 mm. The SCA model has 32 clusters in the matrixphase (k₁=32).

FIG. 52 shows load-displacement curves predicted by SCA databases withdifferent number clusters k₁ before energy regularization according toembodiments of the invention.

FIG. 53 shows load-displacement curves predicted by SCA databases withdifferent number clusters k₁ after energy regularization according toembodiments of the invention.

FIG. 54 shows crack patterns predicted by SCA databases with differentnumber clusters k₁ after energy regularization. according to embodimentsof the invention.

FIG. 55 shows geometry of the 3D double-notched coupon and microscaleSCA model with 8 clusters in the matrix phase according to embodimentsof the invention. The non-local length parameter l₀ is equal to 2 mm.

FIG. 56 shows macroscale FE mesh with local refinement (left) andload-displacement curve of the 3D concurrent simulations with hardinclusions (right) according to embodiments of the invention.

FIG. 57 shows the von Mises stress distributions and crack patterns ofthe 3D concurrent simulations at two loading states according toembodiments of the invention. The microscale stress and damage fieldsinside the RVE are shown in sequence at three different integrationpoints marked in the macroscale model.

DETAILED DESCRIPTION OF THE INVENTION

The present invention will now be described more fully hereinafter withreference to the accompanying drawings, in which exemplary embodimentsof the present invention are shown. The present invention may, however,be embodied in many different forms and should not be construed aslimited to the embodiments set forth herein. Rather, these embodimentsare provided so that this disclosure will be thorough and complete, andwill fully convey the scope of the invention to those skilled in theart. Like reference numerals refer to like elements throughout.

The terms used in this specification generally have their ordinarymeanings in the art, within the context of the invention, and in thespecific context where each term is used. Certain terms that are used todescribe the invention are discussed below, or elsewhere in thespecification, to provide additional guidance to the practitionerregarding the description of the invention. For convenience, certainterms may be highlighted, for example using italics and/or quotationmarks. The use of highlighting and/or capital letters has no influenceon the scope and meaning of a term; the scope and meaning of a term arethe same, in the same context, whether or not it is highlighted and/orin capital letters. It will be appreciated that the same thing can besaid in more than one way. Consequently, alternative language andsynonyms may be used for any one or more of the terms discussed herein,nor is any special significance to be placed upon whether or not a termis elaborated or discussed herein. Synonyms for certain terms areprovided. A recital of one or more synonyms does not exclude the use ofother synonyms. The use of examples anywhere in this specification,including examples of any terms discussed herein, is illustrative onlyand in no way limits the scope and meaning of the invention or of anyexemplified term. Likewise, the invention is not limited to variousembodiments given in this specification.

It will be understood that, although the terms first, second, third,etc. may be used herein to describe various elements, components,regions, layers and/or sections, these elements, components, regions,layers and/or sections should not be limited by these terms. These termsare only used to distinguish one element, component, region, layer orsection from another element, component, region, layer or section. Thus,a first element, component, region, layer or section discussed below canbe termed a second element, component, region, layer or section withoutdeparting from the teachings of the present invention.

It will be understood that, as used in the description herein andthroughout the claims that follow, the meaning of “a”, “an”, and “the”includes plural reference unless the context clearly dictates otherwise.Also, it will be understood that when an element is referred to as being“on,” “attached” to, “connected” to, “coupled” with, “contacting,” etc.,another element, it can be directly on, attached to, connected to,coupled with or contacting the other element or intervening elements mayalso be present. In contrast, when an element is referred to as being,for example, “directly on,” “directly attached” to, “directly connected”to, “directly coupled” with or “directly contacting” another element,there are no intervening elements present. It will also be appreciatedby those of skill in the art that references to a structure or featurethat is disposed “adjacent” to another feature may have portions thatoverlap or underlie the adjacent feature.

It will be further understood that the terms “comprises” and/or“comprising,” or “includes” and/or “including” or “has” and/or “having”when used in this specification specify the presence of stated features,regions, integers, steps, operations, elements, and/or components, butdo not preclude the presence or addition of one or more other features,regions, integers, steps, operations, elements, components, and/orgroups thereof.

Furthermore, relative terms, such as “lower” or “bottom” and “upper” or“top,” may be used herein to describe one element's relationship toanother element as illustrated in the figures. It will be understoodthat relative terms are intended to encompass different orientations ofthe device in addition to the orientation shown in the figures. Forexample, if the device in one of the figures is turned over, elementsdescribed as being on the “lower” side of other elements would then beoriented on the “upper” sides of the other elements. The exemplary term“lower” can, therefore, encompass both an orientation of lower andupper, depending on the particular orientation of the figure. Similarly,if the device in one of the figures is turned over, elements describedas “below” or “beneath” other elements would then be oriented “above”the other elements. The exemplary terms “below” or “beneath” can,therefore, encompass both an orientation of above and below.

Unless otherwise defined, all terms (including technical and scientificterms) used herein have the same meaning as commonly understood by oneof ordinary skill in the art to which the present invention belongs. Itwill be further understood that terms, such as those defined in commonlyused dictionaries, should be interpreted as having a meaning that isconsistent with their meaning in the context of the relevant art and thepresent disclosure, and will not be interpreted in an idealized oroverly formal sense unless expressly so defined herein.

As used in this disclosure, “around”, “about”, “approximately” or“substantially” shall generally mean within 20 percent, preferablywithin 10 percent, and more preferably within 5 percent of a given valueor range. Numerical quantities given herein are approximate, meaningthat the term “around”, “about”, “approximately” or “substantially” canbe inferred if not expressly stated.

As used in this disclosure, the phrase “at least one of A, B, and C”should be construed to mean a logical (A or B or C), using anon-exclusive logical OR. As used herein, the term “and/or” includes anyand all combinations of one or more of the associated listed items.

The methods and systems will be described in the following detaileddescription and illustrated in the accompanying drawings by variousblocks, components, circuits, processes, algorithms, etc. (collectivelyreferred as “members”). These members may be implemented usingelectronic hardware, computer software, or any combination thereof.Whether such elements are implemented as hardware or software dependsupon the particular application and design constraints imposed on theoverall system. By way of example, a member, or any portion of anmember, or any combination of members may be implemented as a“processing system” that includes one or more processors. Examples ofprocessors include microprocessors, microcontrollers, graphicsprocessing units (GPUs), central processing units (CPUs), applicationprocessors, digital signal processors (DSPs), reduced instruction setcomputing (RISC) processors, systems on a chip (SoC), basebandprocessors, field programmable gate arrays (FPGAs), programmable logicdevices (PLDs), state machines, gated logic, discrete hardware circuits,and other suitable hardware configured to perform the variousfunctionality described throughout this disclosure. One or moreprocessors in the processing system may execute software. Software shallbe construed broadly to mean instructions, instruction sets, code, codesegments, program code, programs, subprograms, software components,applications, software applications, software packages, routines,subroutines, objects, executables, threads of execution, procedures,functions, etc., whether referred to as software, firmware, middleware,microcode, hardware description language, or otherwise.

Accordingly, in one or more example embodiments, the functions describedmay be implemented in hardware, software, or any combination thereof. Ifimplemented in software, the functions may be stored on or encoded asone or more instructions or code on a computer-readable medium.Computer-readable media includes computer storage media. Storage mediamay be any available media that can be accessed by a computer. By way ofexample, and not limitation, such computer-readable media can comprise arandom-access memory (RAM), a read-only memory, an electrically erasableprogrammable read-only memory (EEPROM), optical disk storage, magneticdisk storage, other magnetic storage devices, combinations of theaforementioned types of computer-readable media, or any other mediumthat can be used to store computer executable code in the form ofinstructions or data structures that can be accessed by a computer.

The description below is merely illustrative in nature and is in no wayintended to limit the invention, its application, or uses. The broadteachings of the invention can be implemented in a variety of forms.Therefore, while this invention includes particular examples, the truescope of the invention should not be so limited since othermodifications will become apparent upon a study of the drawings, thespecification, and the following claims. For purposes of clarity, thesame reference numbers will be used in the drawings to identify similarelements. It should be understood that one or more steps within a methodmay be executed in different order (or concurrently) without alteringthe principles of the invention.

One aspect of the invention discloses an integratedprocess-structure-property modeling framework for design optimizationand/or performance prediction of a material system. Referring to FIGS.1A-1B, the one exemplary embodiment, the framework includes a database;and a plurality of models coupled with the database. The plurality ofmodels includes process models and mechanical response models. Eachmodel is coupled to the next by passing output information to asubsequent input processor. The process models comprise a powderspreading model, a powder melting model and a grain growth model. Themechanical response models comprise a self-consistent clusteringanalysis (SCA) model, a finite element method (FEM) model and/or afatigue prediction model.

The powder spreading model and the powder melting model operablyexchange geometry information of a powder bed using a stereolithography(STL)-format surface mesh representation. In one embodiment, the powderspreading model is a discrete element method (DEM) powder spreadingmodel.

The powder melting model operably predicts a temperature profile in eachvolume of interest (VOI) with a powder melting simulation. Thetemperature profile is written to the database in a flat file format,and a void space is set to an ambient temperature to distinguish saidvoid space from a dense material. In one embodiment, the powder meltingmodel is a computational fluid dynamics (CFD) powder melting model.

The grain growth model operably accesses the temperature profile andvoid information from the database, and predicts grain information basedon the temperature profile and void information. The grain informationis written to the database along with the voids. The dense material andthe voids are identified based on the temperature profile, and the voidsand free surfaces are thereby included in the grain growth model. In oneembodiment, the grain growth model is a cellular automaton (CA) graingrowth model. In one embodiment, the grain information comprises phaseand orientation.

The SCA and/or FEM models are used, based on the grain and voidinformation provided at each point via the database, to predictstress-strain responses. The stress-strain responses are written to thedatabase. In one embodiment, the SCA model is an SCA crystal plasticitymodel.

The fatigue prediction model operably accesses the stress-strainresponses at each point via the database, and computes a non-localpotency estimate based on the stress-strain responses. The fatigue lifeof the highest potency location is computed to provide a scalar materialresponse metric.

In one embodiment, in operation, the powder spreading model generates aparticle packing configuration within one powder layer with a particlespreading simulation, outputs the particle packing configuration as afirst STL file, and feeds the particle packing configuration into thepowder melting model, thereby reproducing powder spreading and melting;and the powder melting model predicts a solidified shape with the powdermelting simulation, outputs the solidified shape as a second STL file,and feeds the solidified shape into the powder spreading model to applya new powder layer, thereby reproducing powder spreading. In oneembodiment, the manufacturing process of multiple layers and trackscomprises repeating said two simulations and the data exchange betweenthe powder spreading model and the powder melting model.

In one embodiment, in operation, the powder melting model feeds thetemperature profiles and void information to the grain growth model viathe database, and the grain growth model feeds the grain and voidinformation to the SCA model via the database.

In one embodiment, for the mechanical response models, a sub-domain ofparticular interest of the full VOI for grain growth is selected as aninput, the grain and void information provided at each point via thedatabase is assigned as the center point of each voxel in a regular,uniform, cuboid mesh, and said mesh is used for the mechanical responsemodels.

In one embodiment, a material region is discretized by a set of cubiccells, and the temperature history of each cell is determined from athermal-CFD simulation using a linear interpolation.

Another aspect of the invention discloses a method of integratedprocess-structure-property modeling for design optimization and/orperformance prediction of a material system. In one embodiment, themethod includes exchanging geometry information of a powder bed betweena powder spreading model and a powder melting model using a STL-formatsurface mesh representation; predicting a temperature profile in eachVOI by the powder melting model with a powder melting simulation;writing the temperature profile to the database in a flat file format;and setting a void space to an ambient temperature to distinguish saidvoid space from a dense material; accessing the temperature profile andvoid information from the database and predicting grain information by agrain growth model based on the temperature profile and voidinformation; writing the grain information to the database along withthe voids; identifying the dense material and the voids on thetemperature profile, thereby including the voids and free surfaces inthe grain growth model; predicting stress-strain responses by the SCAand/or FEM models based on the grain and void information provided ateach point via the database; and writing the stress-strain responses tothe database; and accessing the stress-strain responses at each pointvia the database and computing a non-local potency estimate by a fatigueprediction model based on the stress-strain responses, wherein thefatigue life of the highest potency location is computed to provide ascalar material response metric.

In one embodiment, said exchanging the geometry information comprises,by the powder spreading model, generating a particle packingconfiguration within one powder layer with a particle spreadingsimulation, outputing the particle packing configuration as a first STLfile, and feeding the particle packing configuration into the powdermelting model, thereby reproducing powder spreading and melting; and, bythe powder melting model, predicting a solidified shape with a powdermelting simulation, outputing the solidified shape as a second STL file,and feeding the solidified shape into the powder spreading model toapply a new powder layer, thereby reproducing powder spreading.

In one embodiment, said exchanging the geometry information furthercomprises repeating said two simulations and the data exchange betweenthe powder spreading model and the powder melting model, for formingmultiple layers and tracks.

In one embodiment, the particle packing configuration is generated usinga discrete element method (DEM).

In one embodiment, the powder melting model is a computational fluiddynamics (CFD) powder melting model.

In one embodiment, the grain growth model is a cellular automaton (CA)grain growth model.

In one embodiment, the SCA model is an SCA crystal plasticity model.

Yet another aspect of the invention discloses a non-transitory tangiblecomputer-readable medium storing instructions which, when executed byone or more processors, cause a system to perform the above methods ofintegrated process-structure-property modeling for design optimizationand/or performance prediction of a material system.

A further aspect of the invention discloses a computational system fordesign optimization and/or performance prediction of a material system,which includes one or more computing devices comprising one or moreprocessors; and a non-transitory tangible computer-readable mediumstoring instructions which, when executed by the one or more processors,cause the one or more computing devices to perform the above methods ofintegrated process-structure-property modeling for design optimizationand/or performance prediction of a material system.

In one aspect, the invention relates to an integratedprocess-structure-property modeling framework for design optimizationand/or performance prediction of a material system, comprising a powderspreading model using the DEM to generate a powder bed; a thermal-fluidflow model of the powder melting process to predict voids andtemperature profile; a CA model to simulate grain growth based on thetemperature profile; and a reduced-order micromechanics model to predictmechanical properties and fatigue resistance of resultant structures byresolving the voids and grains.

In one embodiment, in the CA, a material region is discretized by a setof cubic cells, and the temperature history of each cell is determinedfrom a thermal-CFD simulation using a linear interpolation.

In one embodiment, each model incorporates basic material informationand data provided directly from previous models in the framework topredict the mechanical response and estimated fatigue life of criticalmicrostructures given machine-relevant processing conditions.

In another aspect, the invention relates to method for implementation ofmultiresolution continuum theory (MCT). In one embodiment, the methodincludes decomposing deformation across different microstructural lengthscales, by an MCT model; wherein the macroscale deformation is describedby conventional degrees of freedom and extra degrees of freedom areintroduced to describe the deformation at a microscale; employing amodified homogenization-based Gurson (GTN) model at the macroscale toresult in equations for a void volume fraction, an effective plasticstrain rate, and a matrix flow stress, wherein an additive decompositionof a rate of deformation tensor into elastic and plastic parts allowsfor description of an objective rate of Cauchy stress; solving theresulting equations at each scale using an iterative Newton-Raphsonscheme to update the void volume fraction, the effective plastic strainrate, and the matrix flow stress based on a new deformation tensor,wherein the homogenized void growth behavior is taken to represent theresponse of the primary population of voids, including those initiatedat inclusions, as the material is deformed; and constructing a plasticpotential using microstress and microstress couple at the microscale,wherein a deviatoric strain rate is defined in terms of a relativedeformation between the microscale and the macroscale, and solvingequations of the plastic potential to obtain stress-strain profiles.

In one embodiment, the material structure and the deformation field areresolved at each scale; the resulting internal power is a multi-fieldexpression with contributions from the average deformation at eachscale; and the deformation behavior at each scale is found by testingthe micromechanical response.

In one embodiment, as shown in FIG. 14, the MCT model is implementedwith a user element subroutine including a FEA solver; user materialssubroutines for micro and macro domains; an auxiliary functionsubroutine containing functions used by the user materials subroutinesneeded for the user element subroutine, wherein the auxiliary functionsubroutine is callable by the user material subroutines to finish theupdate of the stress and strain; a force subroutine for calculating theinternal force by calling the user materials subroutines and theauxiliary function subroutine; a parameter subroutine providing inputparameters for the user element subroutine; and a mass subroutine forcomputing a mass matrix for the user element subroutine, wherein theforce subroutine, parameter subroutine and the mass subroutines arecalled by the user element subroutine for solving the equations at eachscale. In one embodiment, the auxiliary function subroutine comprisesGauss integration, computation of inertia and subroutines to calculatethe strain and strain rate. In one embodiment, the input parametersinclude number of micro scales, number of Gauss points and number ofhistory variables.

In one embodiment, the method is applicable to prediction of formationand propagation of a shear band in a material system.

In one embodiment, the method is applicable to simulation of high speedmetal cutting (HSMC) in a material system.

In another aspect, the invention relates to a non-transitory tangiblecomputer-readable medium storing instructions which, when executed byone or more processors, cause a system to perform the above disclosedmethods for implementation of the MCT.

In yet another aspect, the invention relates to a computational systemfor design optimization and/or performance prediction of a materialsystem, comprising one or more computing devices comprising one or moreprocessors; and a non-transitory tangible computer-readable mediumstoring instructions which, when executed by the one or more processors,cause the one or more computing devices to perform above disclosedmethods for implementation of the MCT.

In a further aspect, the invention relates to a method of data-drivenmechanistic modeling of a system for predicting high cycle fatigue (HCF)crack incubation life. In one embodiment, the method comprisesgenerating a representation of the system with matrix, inclusion andvoid, wherein the representation comprises microstructure volumeelements (MVE) that are of building blocks of the system; obtainingstrain contours in the matrix from a linear elastic analysis over apredefined set of boundary conditions; generating clusters from a strainsolution by assigning voxels with similar strain concentration tensorwithin the representation of the material system to one of the clusterswith a unique ID, using k-means clustering; computing a plastic shearstrain field of the material system by solving the Lippmann-Schwingerequation using the generated clusters with a crystal plasticity (CP)model; and predicting the fatigue crack incubation life of the systemusing a fatigue indicating parameter (FIP) based on the plastic shearstrain field.

In one aspect, the invention relates to a non-transitory tangiblecomputer-readable medium storing instructions which, when executed byone or more processors, cause a system to perform the above discloseddata-driven mechanistic modeling methods for predicting high cyclefatigue (HCF) crack incubation life.

In another aspect, the invention relates to computational system fordesign optimization and/or performance prediction of a material system,comprising one or more computing devices comprising one or moreprocessors; and a non-transitory tangible computer-readable mediumstoring instructions which, when executed by the one or more processors,cause the one or more computing devices to perform the above discloseddata-driven mechanistic modeling methods for predicting high cyclefatigue (HCF) crack incubation life.

Without intent to limit the scope of the invention, examples accordingto the embodiments of the present invention are given below. Note thattitles or subtitles may be used in the examples for convenience of areader, which in no way should limit the scope of the invention.Moreover, certain theories are proposed and disclosed herein; however,in no way they, whether they are right or wrong, should limit the scopeof the invention so long as the invention is practiced according to theinvention without regard for any particular theory or scheme of action.

Example 1 Integrated Process-Structure-Property Modeling Framework forAdditive Manufacturing

One objective of modeling for Additive Manufacturing (AM) is to predictthe resultant mechanical properties from given manufacturing parametersand intrinsic material properties, thereby reducing uncertainty in thematerial built. This can dramatically reduce the time and cost for thedevelopment of new products using AM. In this exemplary study, werealize the seamless linking of models for the manufacturing process,material structure formation, and mechanical response through anintegrated multi-physics modeling framework. The sequentially coupledmodeling framework relies on the concept that the results from eachmodel used in the framework are contained in space-filling volumeelements using a prescribed structure. The implementation of theframework demonstrated herein includes: (1) a powder spreading modelusing the discrete element method (DEM) to generate the powder bed and athermal-fluid flow model of the powder melting process to predict voidsand temperature profile, (2) a cellular automaton (CA) model to simulategrain growth based on the temperature profile, and (3) a reduced-ordermicromechanics model to predict the mechanical properties and fatigueresistance of the resultant structures by resolving voids and grains.The simulation results demonstrate qualitative agreement withexperimental observations from literature, showing the appealingpotential of the integrated framework.

Additive Manufacturing (am) Modeling Framework

Here we present a process-structure-property prediction framework formetal additive manufacturing, with an implementation for selectiveelectron beam melting (SEBM) of the commonly used titanium alloy,Ti-6Al-4V. In this framework shown schematically in FIG. 1, each modelincorporates basic material information and data provided directly fromprevious models in the framework to predict the mechanical response andestimated fatigue life of critical microstructures givenmachine-relevant processing conditions. The key idea of the framework isthat structural information, data that can be represented in space usinga collection of volume-filling elements, sufficiently describes thesystem and can thus intermediate models: a simple yet powerful premise.

In an abstract form, the framework includes various “modules” that areaggregated into “hubs”; there may be multiple hubs, each of whichcollects and passes a complete solution for one stage (e.g., processsimulation) to the next stage (e.g., mechanical response prediction).One might think of the modules as discrete processing units capturing aunique facet of the system and the hubs as data management facilitiesfor database queries. To systematize the method we group modules intohubs for (1) material processing and (2) material response.

Each module is stand-alone, designed to answer significant scientificand/or engineering questions independent of the connecting framework. Assuch, the models that make up the modules are only discussed withsufficient detail to elucidate their purpose: the key points of eachmodel are provided. A detailed description of the frameworkimplementation is given below. The process is captured by models forpowder spreading, heat source and paired thermal-fluid modeling, andgrain growth during solidification. The mechanical response predictionincludes reduced order modeling, constitutive law, and performancemetric. The example results are shown with output and discussion foreach model. The specific models are selected to capturemicrostructure-level variations caused by the AM process and the impactof these variations on mechanical properties; these choices are notunique and simply elucidate the framework.

Framework Implementation

FIG. 1A illustrates a schematic diagram of the module-hub frameworkconceptual layout, with images of our current implementationdemonstrating the modules: a selective electron beam melting processwith microstructure-based fatigue life assessment. The data used by theframework at each point in space is described by spatial coordinates andany relevant field data for the current working modules, as showngraphically by a schematic of a representative voxel. The cellularautomata model is shaded gray to indicate that results both with andwithout it are given, to demonstrate the flexibility of the framework.The framework depicted in FIG. 1A contains mechanisms for input andoutput handling as well as intermediate storage. The solution domain isrepresented within a fixed format, flat-file database as a regular gridof volume filling points, with each point described by coordinates (x,y, z location) to identify the unique record, each of which has fieldsthat contain a datum: temperature at a point in time, phase,orientation, various stress and strain tensors expressed componentwise,and a fatigue indicating parameter (FIP) in this implementation. Thesefield values are chosen to match the information required by all of themodels used, and are not necessarily unique or optimal in their abilityto describe the process. FIG. 2 outlines the database format of theflat-file database used to store record data for the framework. Thefields are abbreviated to more clearly show the structure of thedatabase.

This integrated framework is mainly based on one-way coupling asillustrated in FIG. 1B, which is relatively straightforward to implementand test. Each model is directly connected to the next by passing outputinformation to a subsequent input processor: the computational fluiddynamics (CFD) powder melting model feeds temperature profiles and voidsinformation to the CA grain growth model through the flat file database,and the CA grain growth model feeds the grains and void information tothe self-consistent clustering analysis (SCA) crystal plasticity model,also through the flat file database. In this case, data is promulgatedin predominately one direction (process to structure to properties): a“backwards” coupling (e.g., structure to process) is not considered.Latent heat is considered an effective simplification of the influenceof microstructure evolution on the temperature field and molten poolflow. Previous thermal or thermal-fluid models with this simplificationhave shown simulated temperature history and molten pool behavior ingood agreement with experiments. The grain growth models have also shownqualitative agreement with experimental results using simple one-waycoupling that does not incorporate the influence of mechanicaldeformation (e.g., thermal stress) on grain growth. Therefore, thissequentially coupled framework should be sufficient to predict thetemperature profiles, material structures and mechanical behaviors. Froma top-down approach as illustrated below, this framework can be used toderive process-structure-property relationships, while a bottom-upapproach could be used to solve design and inverse problems.

In certain embodiments, the framework proceeds as follows:

-   -   The powder spreading model and powder melting model exchange        high-resolution geometry information of the powder bed, using a        stereolithography (STL)-format surface mesh representation. That        is, the DEM powder spreading model generates the particle        packing configuration within one powder layer, outputs as an STL        file, and feeds into the CFD powder melting, thereby reproducing        powder spreading and melting, as illustrated by the black arrows        in FIG. 1B; the CFD powder melting model predicts the solidified        shape, outputs as an STL file, and feeds into the DEM powder        spreading model to apply a new powder layer, thereby reproducing        powder spreading, as illustrated by the gray arrows in FIG. 1B.        The manufacturing process of multiple layers and tracks can be        represented by repeating these two simulations and the data        exchange between them.    -   The temperature profile in the volume of interest (VOI) is        predicted by the powder melting model and written to a database        in the flat file format shown in FIG. 2. By convention, void        space is set to an ambient temperature to distinguish it from a        dense material.    -   The grain growth model reads the temperature field history from        the database, and interpolates linearly to match the grid        spacing when needed. The dense material and voids are identified        based on the temperature. Voids and free surfaces are thereby        included in the grain growth model.    -   The grain structure (phase and orientation), as predicted by the        grain growth model, is added to the database along with the        voids.    -   For the mechanical response models, a sub-domain of particular        interest, e.g., the largest void, of the full VOI for grain        growth is selected as the input; the grain and void information        provided at each point via the database are interpreted as the        center point of each voxel in a regular, uniform, cuboid mesh.        This mesh can be used for all three mechanical response modules:        FEM, SCA, and fatigue prediction.    -   The SCA and FEM models are used to predict stress-strain        response, which is added to the database.    -   The fatigue prediction model reads the stress-strain behavior at        each point, and computes a non-local potency estimate. The        fatigue life of the highest potency location is computed to        provide a scalar material response metric.

Process Modeling Hub

The first hub codifies the process models. Inputs include processconditions and material information. Process conditions incorporate amultitude of factors, such as the scan pattern, input power, powder bedthickness, and build chamber conditions (e.g., temperature, degree ofvacuum).

In the relatively simple case shown here, the scan pattern is atwo-pass, two-layer Z configuration: each layer is added using the samepath at the new height. A diagram of the scan pattern is shown in FIG.3. Between layers the powder spreading model is used to place the newbed of powder particles. The temperature history of each point and thefinal geometry of the CFD powder melting simulation are outputinformation, which are subsequently used as the input to the graingrowth model. The grain growth model input processor identifies thevoids and predicts the grain structures based on thermal informationfrom the CFD model. The output of this hub is the overall microstructureinformation, crystallography and geometry, predicted with thethermal-CFD model and grain growth model.

A continuum-based thermal model using the Finite Element Method (FEM),for example, could replace the thermal predictions of the high-fidelitythermal-CFD model used here. In this study, the high-fidelity models forpowder spreading and melting are intended to provide a relativelyaccurate description of the thermal conditions and void informationcompared to the simpler models required for part-scale analysis. Thesehigh-fidelity models have high computational cost, limiting the totalvolume of material that can be investigated in the following examples.

Powder Spreading

The arrangement of particles in the powder bed is determined with aparticle spreading simulation. The distribution and dynamics ofparticles within the bed or on a substrate can influence the final buildcondition, e.g., through surface roughness or defect formation betweentracks or layers. Thus, spreading is simulated with accurate processvariables such as the rake geometry, substrate geometry, layerthickness, powder size distribution, and frictional contact/collisionsbetween particles and with the rake. In this case, particles arespherical, and have diameters following a Gaussian distribution between30 and 50 microns. A collection of particles is generated, dropped onthe previous layer, and spread with a rake. A surface mesh, stored in anSTL-format file, represents the resulting geometry. The parameters usedin for the module are listed in Table 1-1, and material parameters ofTi-6Al-4V for the powder spreading model are listed in Table 1-2.

TABLE 1-1 Input and output for powder spreading model Input Particlesize distribution Spreading tool geometry/description Substrateshape/material Output Powder bed state (STL geometry) after spreading

TABLE 1-2 Material parameters of Ti—6Al—4V for the powder spreadingmodel Property Value Density (ρ) 4000 kg/m³ Young's modulus (E) 124 GPaPoisson ration (ν)  0.41 Sliding friction coefficient (μ_(f)) 0.5Restitution coefficient (θ) 0.5 Surface energy density (ψ) 0.0002 J/m²

Powder Melting

The powder geometry is passed to a coupled thermal-fluid solver tocompute thermal response and material redistribution during melting andsubsequent re-solidification. The governing equations are continuity,momentum conservation and energy conservation, given by

$\begin{matrix}\left\{ {\begin{matrix}{{{\nabla{\cdot \left( {\rho \; v} \right)}} = 0},} \\{{{{\frac{\partial}{\partial t}\left( {\rho \; v} \right)} + {\nabla{\cdot \left( {\rho \; {v \otimes v}} \right)}}} = {{\nabla{\cdot \left( {\mu {\nabla v}} \right)}} - {\nabla p} + {\rho \; g} + f_{B}}},} \\{{{\frac{\partial}{\partial t}\left( {\rho \; h} \right)} + {\nabla{\cdot \left( {\rho \; {vh}} \right)}}} = {q + {\nabla{\cdot \left( {k{\nabla T}} \right)}}}}\end{matrix},} \right. & \left( {1\text{-}1} \right)\end{matrix}$

where flow is assumed to be incompressible, laminar, and Newtonian; inthe momentum conservation equation, the major driving forces areincorporated, including gravity (g), buoyancy (f_(B), Boussinesqapproximation), viscosity (μ), recoil pressure, surface tension andMarangoni forces. The solid phase is approximated as a fluid withextremely large viscosity. Input energy (q) from an electron beam, heatconduction (k, the thermal conductivity), and latent heat of phasechange (captured in the definition of enthalpy h) are incorporated inthe energy conservation equation. Surface convection is obviated due tothe evacuated environment. An important term is that of the heat source(q), representing an electron beam impinging upon the bed of particles.A volumetric heat source model has been derived based on nano-scalesimulations of electron-atom interactions in our previous study, and isgiven by:

q=ηQF _(section) F _(pene)  (1-2)

where q is the absorbed energy density, F_(section) is the absorbedenergy distribution within the beam cross-section, F_(pene) is theabsorbed energy distribution along the penetration depth, η is theenergy absorptivity, and the beam power Q is the product of theacceleration voltage and beam current. The results shown in thefollowing sections are for Case 1 in Table 1-3. The absorptivity, η,depends strongly upon the local incident angle and thus varies widely.

The fully coupled thermo-fluid flow equations governing the evolution ofpowder particles are computed with a finite volume method (FVM). Withinthis, the free surface is tracked using the volume of fluid (VOF)method,

$\begin{matrix}{{\frac{\partial F}{\partial t} + {\nabla{\cdot ({Fv})}}} = 0.} & \left( {1\text{-}3} \right)\end{matrix}$

where F represents the fluid volume fraction. At each time step, thefree surface is reconstructed with VOF, the heat source in eachcomputational cell is applied using the heat source model describedabove, the effects of evaporative mass loss, energy loss and recoilpressure on the free surface are computed on each cell, and the freesurface is updated. For these simulations, the material parameters aregiven in Table 1-4. A list of the inputs and outputs of this module isprovided in Table 1-5.

The cell size for this model is 5 μm: sufficient to eliminate meshsensitivity, based on a previous study. The computation time for the2-layer, 2-track case (see FIG. 3) is around 700 hours on a desktopcomputer using an Intel Core i7-2600 CPU.

The temperature history of each computational cell is output into theflat-file database for use in the Cellular Automata (CA) model. Todistinguish between a solid/liquid domain and void space, i.e., abovethe build surface and any voids within the material, void areas areassigned ambient temperature, while the temperature of the solid/liquiddomain is no lower than the initial temperature (about 900K) due to thepreheating procedure in the SEBM.

TABLE 1-3 Build parameters for Cases 1 and 2 Parameter Case 1 Case 2Hatch spacing 200 μm 240 μm Laser power 60 W 60 W Hatch strategyZ-pattern Z-pattern Scan speed 0.5 m/s 0.5 m/s

TABLE 1-4 Thermal and flow properties of Ti—6Al—4V Property ValueDensity (ρ) 4000 kg/m³ Solidus temperature (T_(s)) 1878 K Liquidustemperature (T_(l)) 1928 K Latent heat of melting (L_(m)) 2.86 × 10⁵J/kg Latent heat of evaporation (L_(v)) 9.7 × 10⁶ J/kg Saturated vaporpressure (P_(s0)) at 1.013 × 10⁵ Pa T₀ = 3315 K Specific heat (c) 872J/(K · kg) Thermal conductivity at solidus (k) 16 W/(m · K) Thermalconductivity at liquidus (k) 32 W/(m · K) Surface radiation coefficient(α_(b)) 0.4 Surface tension coefficient (σ) 1.68 N/m Temperaturesensitivity of σ (σ_(s) ^(T)) 0.00026 Viscosity (μ) 0.005 Pa · s Domainsize 0.75 mm × 0.225 mm × 0.055 mm Mesh size 150 × 45 × 10 Cell size0.005 mm

TABLE 1-5 Input and output for thermal-CFD model Input Powder bed stateThermal processing conditions Material properties Output Thermal historyof each point in the volume Geometry (including voids)

Grain Growth

Grain formation during solidification may be computed from thetemperature field, provided by the thermal-CFD (Computational FluidDynamics) module with one-way coupling. The CA model is used in thisstudy. In the CA, the material region is discretized by a set of cubiccells. The temperature history of each cell is determined from thethermal-CFD simulation using a linear interpolation. The input andoutput of this module is summarized in Table 1-6.

TABLE 1-6 Input and output for grain growth model Input Thermal historyat each point Material properties, parameters Output Phase ID (includesvoid phase) Grain orientation (Bunge Euler angles)

There are two main sub-models required for the CA. The first, aheterogeneous nucleation model, includes defining the distribution ofnucleation sites, the critical undercooling value (the temperature atwhich nucleation occurs at each site), and the crystal orientation ofnewly nucleated grains. In this CA model, nucleation occurring at theboundary of a liquid volume is treated differently from nucleationwithin the bulk of the liquid volume through a “wall” nucleation densityn_(max,w) with units of m⁻² and a “liquid,” or bulk, nucleation densityn_(max,l) with units of m⁻³. Prior to the simulation, the total numberof nucleation sites at the surface and in the bulk are calculated andassigned critical undercooling values. This critical undercooling variesfrom site to site; the variation is assumed to follow a Gaussiandistribution (given by the values of the mean, ΔT, and standardderivation, ΔT_(σ)). The surface and bulk nucleation sites may usedifferent values of ΔT and ΔT_(σ), distinguished by subscripts w and l.

The second sub-model, a model of grain growth during the solidificationprocess, controls the rate and direction for which existing grains growby assimilating surrounding cells. For a newly nucleated grain, theinitial dendritic network within the cell is assumed to be a regularoctahedral envelope bounded by

111

planes and to have a random crystallographic orientation defined by aset of Euler angles. The six half-diagonals of the octahedron correspondto the primary dendritic growth directions of the grain. Grain growth issimulated by extending these half-diagonals based on the dendrite tipvelocity, v(ΔT). Here ΔT is the local undercooling at the center of thecell that owns the envelope; ownership is determined by a decenteredoctahedron growth algorithm. As time proceeds, the envelope grows andeventually engulfs neighboring cells to propagate the grain. For anactive envelope located in a cell, e.g., cell i, the change in length ofits diagonal, ΔL_(i), is calculated as

ΔL _(i) =v(ΔT _(i))δt  (1-4)

where ΔT_(i) is the total undercooling at the center of the cell and δtis the current time step. The growth rate, v(ΔT_(i)), is usuallyestimated based on the undercooling by a polynomial formulation. We use

v(ΔT)=a ₁ ΔT+a ₂ ΔT ² +a ₃ ΔT ³,  (1-5)

where a₁, a₂ and a₃ are fit parameters calibrated to experimental ortheoretical results based on the dendrite tip growth kinetics. Theparameters for this portion of hub 1 are provided in Table 1-7.

The conditions of AM (high temperature gradient and cooling rate) resultin grain growth dominated by epitaxial growth from existing grainssurrounding the melt pool. Thus, the grain structure of the substrateand metal powders are obtained first through the nucleation parametersshown in Table 1-7 before the melting process; during the melting andre-solidification process, nucleation is assumed to progress only fromthe solid material boundary to better capture growth conditions underAM. The prediction results from the CA model are written to the databaseas a phase ID (a unique integer) and a predicted crystallographicorientation (Euler angle triplet following the Bunge convention) foreach voxel in the CA domain. This database can be read directly toconstruct a voxel mesh, since material properties are associated withphase ID and orientation.

TABLE 1-7 CA model parameters; material parameters Parameter(s) Value(s)T_(melting) 1928° C. T_(ambient) 298° C. Domain size 0.35 mm × 0.225 mm× 0.055 mm Mesh size 280 × 180 × 44 Cell edge size 1.25 × 10⁻³ mmNucleation, wall: 5 × 10¹⁰ m⁻², 2° C., 0.5° C. n_(max,w), ΔT _(w),ΔT_(w,σ) Nucleation, liquid: 5 × 10¹⁴ m⁻³, 2° C., 0.5° C. n_(max,l), ΔT_(l), ΔT_(l,σ) Growth rate: α₃, α₂, α₁ 0.0, 2.03 × 10⁻⁴ m/(s ° C.²),0.544 × 10⁻⁴ m/(s ° C.)

Mechanical Response Prediction Hub

The second hub codifies our mechanical response and performanceprediction methods. Input to this hub is crystallographic and geometricmicrostructure information, as well as material properties forconstitutive model calibration. The output of this hub is a predictionof mechanical response, i.e., overall stress-strain curves, localplastic strains and stresses, and the high-cycle fatigue incubationlife. All information (phase ID, grain orientation, etc.) is carried byvoxels. A state-of-the-art reduced order modeling scheme calledSelf-consistent Clustering Analysis (SCA) was used to speed up thesimulations. Volumes of interest (VOIs) for computing the fatigueindicating parameter (FIP) and predicting fatigue life are selected fromthe larger thermal-CFD-grain growth region; a subset with known spatialcoordinates can simply be queried from the database, which contains thepredicted microstructure (void size, shape, location, grain size, shape,orientation, etc.). All the input and output data to and from thismodule is listed in Table 1-8.

TABLE 1-8 Input and output for mechanical response model Input Phase ID(including voids) Grain orientations (if not provided assume singlecrystal in weakest orientation) Material properties Boundary/loadingconditions (assumes periodic BCs) Output Stresses Strains

Crystal Plasticity for Material Law

To understand the microscale mechanics, in this case the influence ofunusual grain geometry and the relatively frequent occurrence of voidsin AM, we apply a crystal plasticity (CP) constitutive law. In the CPframework, deformation is described by summing the shear slip rate ofall slip systems in all the crystals under consideration:

{tilde over (L)} ^(p)=Σ_(α=0) ^(N) ^(slip) {dot over (γ)}^((α))({tildeover (s)} ^((α)) ⊗{tilde over (m)} ^((α)))  (1-6)

where {tilde over (L)}^(p) is the plastic velocity gradient in theintermediate configuration; α is the index of a slip system; {dot over(γ)}^((α)) is the rate of shear slip in slip system a; {tilde over(s)}^((α)) is the slip direction of that system and {tilde over(m)}^((α)) is the slip plane normal. In the model used here, a simplerate-dependent power-law with backstress governs the rate of shear slip:

$\begin{matrix}{{\overset{.}{\gamma}}^{(\alpha)} = {{\overset{.}{\gamma}}_{0}{\frac{\tau^{(\alpha)} - a^{(\alpha)}}{\tau_{0}^{(\alpha)}}}^{({m - 1})}\; \left( \frac{\tau^{(\alpha)} - a^{(\alpha)}}{\tau_{0}^{(\alpha)}} \right)}} & \left( {1\text{-}7} \right)\end{matrix}$

where {dot over (γ)}₀ is a reference shear strain rate, τ^((α)) is theresolved shear stress on slip system α, a^((α)) is a backstress forkinematic hardening, and τ₀ ^((α)) is the reference shear strength, theevolution of which depends on the terms for direct hardening and dynamicrecovery in a hardening law. The reference shear strength τ₀ ^((α))evolves based on the expression

{dot over (τ)}₀ ^((α)) =HΣ _(β=1) ^(N) ^(slip) q ^(αβ){dot over(γ)}^((β)) −Rτ ₀ ^((α))Σ_(β=1) ^(N) ^(slip) |{dot over(γ)}^((β))|,  (1-8)

where H is the direct hardening coefficient and R is the dynamicrecovery coefficient and q^(αβ) is the latent hardening ratio given by:

q ^(αβ)=χ+(1−χ)δ_(αβ)  (1-9)

where χ is a latent hardening parameter (1: without latent hardening, 0:with latent hardening). The backstress a^((α)) evolves based on theexpression:

{dot over (a)} ^((α)) =h{dot over (γ)}(α)−ra|{dot over (γ)}^((α))|,  (1-10)

where h and r are the direct hardening and dynamic recovery factorsrespectively.

It has been observed that the microstructure of Ti-6Al-4V produced bySEBM is dominated by hexagonal-close-packed (hcp) α grains which aretransformed from the prior body-centered-cubic (bcc) β grains. The αphases originating from the same parent β grain have similarorientation. For this demonstration of the framework we make the simple,though inaccurate, assumption that each parent β grain, under thermalprocessing, produces a single, aligned child α grain. Slip systems usedfor the dominant α-phase of Ti-6Al-4V are 3 (1120){0001} basal, 3(1120){1010} prismatic, 6(1120){1011} first order pyramidal and12(1123){1011} second order pyramidal. These are shown in Table 1-9.Parameters for the crystal plasticity material model used both foroffline and online stages are also given in Table 1-9, where C_(ij) arethe stiffness components in Voigt notation.

TABLE 1-9 Calibrated elastic and plastic crystal plasticity parametersfor primary α phase. Parameters with a ^(#) are from Thomas et al.(Materials Science and Engineering: A, 553: 164-175, 2012). Others areselected to match experimental results Plasticity: Elasticity: PyramidalPyramidal C* MPa Phase Basal Prismatic

 a 

 c + a 

C₁₁ = C₂₂  1.7 × 10⁵ γ₀ ^(#) s⁻¹ 0.0023 0.0023 0.0023 0.0023 C₃₃ 2.04 ×10⁵ m^(#) 50 50 50 50 C₄₄ = C₅₅ 1.02 × 10⁵ τ₀ MPa 284.00 282.24 395.00623.30 C₆₆ 0.36 × 10⁵ a^(t=0) MPa 0.0 0.0 0.0 0.0 C₁₂ = C₂₁ 0.98 × 10⁵ HMPa 1.0 1.0 1.0 1.0 C₁₃ = C₃₁ 0.86 × 10⁵ R MPa 0.0 0.0 0.0 0.0 C₂₃ = C₃₂0.86 × 10⁵ h MPa 500.0 500.0 500.0 500.0 other C_(ij) 0 r MPa 0.0 0.00.0 0.0 χ 1.0 1.0 1.0 1.0

Self-Consistent Clustering Analysis (SCA)

The boundary value problem of the VOI to be solved can be written in aLagrangian formulation as:

$\begin{matrix}\left\{ \begin{matrix}{{{{div}\; {\sigma (x)}} = 0},\; {\forall{x \in \Omega}}} \\{{\sigma (x)} = {f\left( {x,\epsilon,\overset{.}{\epsilon},s} \right)}} \\{\epsilon = {\epsilon^{*} + \overset{\_}{\epsilon}}}\end{matrix} \right. & \left( {1\text{-}11} \right)\end{matrix}$

where x is a point in the VOI Ω₀; σ(x) is the local stress tensor, whichis a function of the local strain ϵ, its rate {dot over (ϵ)} and someinternal variables s for a general elasto-viscoplastic material. Thelocal strain ϵ is the sum of a fluctuating term ϵ* and a prescribedoverall strain ϵ. Assuming periodic boundary conditions, the aboveboundary value problem is equivalent to the following Lippmann-Schwingerformulation

ϵ=−Γ⁰*(σ−C ⁰:ϵ)+Ē,  (1-12)

where Γ⁰ is the Green's operator corresponding to the referencestiffness C⁰ and * denotes the convolution operation.

The reduced order mechanical modeling scheme used here is based ondiscretization of the Lippmann-Schwinger formulation. The way thisapproach differs from other reduced order methods is that thedecomposition of the VOI is based on knowledge of the strainconcentration. This is used to group regions of material with similardeformation behavior. This information is acquired in the “offline”stage of the SCA approach, where a simple (elastic) direct numericalsimulation of the microstructure of interest is conducted. In the“online” stage, any nonlinear material law and/or loading path can beused.

In this case, the strain concentration tensor of each voxel is obtainedwith a linear elastic finite element model conducted in Abaqus usingC3D8R elements. This model computes the elastic response of a VOI withperiodic boundary conditions under three uniaxial tensile loading casescorresponding to x, y, z directions respectively. The stretch directionis the same as applied during the online stage. The elastic strains thusobtained are used in a k-means clustering algorithm for domaindecomposition, where the squared Euclidean distance measure is used forclustering.

Four fully-reversed unidirectional load cycles at various strainamplitudes with crystal plasticity constitutive law were simulatedduring the online analysis. These cycles result in a stabilized value ofΔγ_(max) ^(p) and a known σ_(max) ^(n), used later to compute a FIP. Afull accounting of the parameters used for both the offline and onlinemodules is provided in Table 1-10.

TABLE 1-10 Offline and online model parameters Parameter(s) Value(s)Domain size 0.05 mm × 0.05 mm × 0.05 mm Mesh size size 40 × 40 × 40Element edge length 1.25 × 10⁻³ mm Material law, offline linearelasticity Material law, online crystal plasticity Strain amplitude,online 0.5%, 0.375%, 0.3% of strain cycles   4 Load ratio, R =ϵ_(min)/ϵ_(max) −1 Strain rate, {dot over (ϵ)} 0.1 s⁻¹

Performance Prediction: Fatigue Life

A performance metric simplifies ranking the predicted build quality of apart or process parameter set to the comparison of a scalar factor. Toprovide a metric, this module estimates the high-cycle fatigue life ofthe predicted microstructure. A Fatemi-Socie FIP appropriate formicrostructural analysis is used, which is given by

$\begin{matrix}{{F\; I\; P} = {\frac{\Delta \; \gamma_{\max}^{p}}{2}\left( {1 + {\kappa \frac{\sigma_{n}^{\max}}{\sigma_{y}}}} \right)}} & \left( {1\text{-}13} \right)\end{matrix}$

where Δγ_(max) ^(p) is the maximum of the cycle-to-cycle change of theplastic shear strain, σ_(n) ^(max) is the stress normal to Δγ_(max)^(p), σ_(y) is the yield stress, and κ is a normal stress factor. Thevalue of Δγ_(max) ^(p) saturates after a few load cycles and thesaturated value can be used to compute TIP and N_(inc). The relationshipbetween FIP and N_(inc) for Ti-6Al-4V is calibrated to experimental highcycle fatigue life data for wrought (nominally defect free) Ti-6Al-4V.Here, this calibration has been done to reduce the total RMS differencebetween the experimental data and the N_(inc) predicted for thepolycrystalline, defect free VOI at 0.5% and 0.375% strain amplitudessimultaneously (this is the simulation condition that best matches thatof the experimental conditions). A moving volume average with windowsize of about 10% of the volume of the microstructure of interest (e.g.,the mean void size) is used to compute the nonlocal FIP. The maximumnonlocal FIP, NFIP_(max), is correlated with fatigue crack incubationlife, N_(inc), through

NFIP_(max)=γ _(f)(2N _(inc))^(c)  (1-14)

where γ _(f) and c, multiplicative and exponential factors, are used tofit experimental high cycle fatigue data. The input and output of themodule are summarized in Table 1-11.

TABLE 1-11 Input and output for fatigue prediction Input StressesStrains Material properties Output FIP N_(inc)

Example Results and Discussion

The final result of this integrated framework is the computedperformance metric. Here we also present intermediate results from eachof the models described above, to make the flow of information betweenmodules and hubs within the framework much clearer by example. Each ofthe modules contains a state-of-the-art method in itself; the importantpoint that we demonstrate, to our knowledge for the first time in theopen literature, is that the modules used to predict processing andmaterial response to mechanical loading during service have beendirectly linked. A comparison of the final metric for build qualitybetween cases with different processing conditions is made. The exampleshere show the influence of process parameter choices on simulatedperformance with two distinct cases shown in Table 1-3: one with hatchspacing 200 μm and the other with hatch spacing 240 μm. This allows usto directly show the differences in that result from different processparameters. The influence can be seen in FIG. 12, where a larger hatchspacing results in the prediction of voids, reducing estimated fatiguelife.

Powder Spreading

The powder spreading model produces a relatively densely packed bed ofparticles, with random spatial distribution. Panel (a) of FIG. 4 showsthe powder arrangement for the first layer, spread on a smooth Ti-6Al-4Vsubstrate. Panel (b) of FIG. 4 shows the arrangement of particles forthe second layer, spread upon the surface resulting from the thermal-CFDprocess simulation of the first layer. The surfaces shown in thesefigures are represented by STL meshes, and given thereby to the firststep of the thermal-CFD code: a 3D mesh generator. In FIG. 4, the firstpowder layer with a thickness of 0.05 mm spread on the substrate, whilethe second powder layer with a thickness of 0.05 mm spread on thesolidified surface of the first layer. The figures are colored by theheight from the substrate surface. In panels (a) and (b), the uppersub-figures show views from above, and the lower ones show views frombelow (the z direction), with the non-powder material hidden.

Powder Melting

The geometry generated by the powder spreading model can be seen aftermelting has been simulated in FIG. 6 for Case 2 in Table 1-3. Note thematch between the unmelted portions of FIG. 6 and the same locations inFIG. 4: the results from one model are used directly in the next. FIG. 5shows an example of the temperature evolution at a single point, wherethe inset curve highlights the kink caused by latent heat. Rapid heatingand cooling with a slight kink caused by the latent heat ofsolidification is predicted; the solidification kink is shown in detailin the inset. The temperature peaks again upon subsequent passes of theelectron beam. This matches the well-documented thermal profile for theSEBM.

The voids predicted by the CFD model are highlighted in the inset inFIG. 6, where void space is colored and the material hidden. These voidsresult from lack of fusion between particles near the boundaries betweenthe melt tracks. The irregular shapes and positions (between tracks andlayers) match qualitatively with lack of fusion voids observedexperimentally. Because no vapor phase is explicitly included in thisCFD model, no voids caused by entrained gas are predicted. Simulatedbuilds with different conditions, e.g., Cases 1 and 2 in Table 1-3,result in different geometric configurations and thermal profiles.

As shown in FIG. 6, voids are observed in the CFD simulation ofmultiple-layer multiple-track manufacturing process. This is a 3D viewof the free surfaces after manufacturing, colored by the height from thesubstrate surface. In FIG. 6, the upper image shows a view from above,and the lower image shows a view from below, with the substrate materialhidden, and the inset image highlights a void predicted during thebuild.

Grain Growth

This model predicts solidification front motion and grain growth giventhe thermal history; Panel (a) of FIG. 7 shows the molten pooltemperature at t=0.157 ms. Panels (a)-(d) of FIG. 7 track theprogression of the solidification front and the subsequent primary phasegrain growth at t=0.234 ms, t=0.438 ms, and t=0.931 ms, where graincoloring is selected to maximize contrast between nearby grains. Theemergence of columnar grains from an existing substrate is captured withqualitative similarity to experimental results which reports the cubicβ-grain structure that forms during initial solidification. The CA modelcaptures the growth of the primary β-phase. Sub-granular structures,such as dendrites, are not considered explicitly and thus cannot beobserved.

Grains appear tilted in the direction of the heat source motion, showinga trend also observed in experiments and other simulations of graingrowth during AM. This is shown by the longitudinal cross section viewof the region of interest in FIG. 8. FIG. 9 shows a transverse crosssection of the region with radial, epitaxial grain growth from theunderlying material.

Mechanical Response Prediction

When run without the CA module, we assume a single crystal oriented suchthat the Schmid factors, the products of the glide plane and glideorientation cosines, are maximized: the weakest orientation,representing a “worst-case” analysis. A more realistic, polycrystallinegrain structure enabled by the CA model allows us to capture some of theeffects of the crystalline texture on the material response. Panels(a)-(b) of FIG. 10 show four polycrystalline VOIs selected from the CAsolution for a two-track, two-layer simulation. Three VOIs are selectedthroughout the height of the build (the black, purple, and greenoutlines) and a fourth VOI (red outline) is selected from the intertrackarea. The fourth region in the CA volume is selected because theintertrack area contains a large void caused by lack-of-fusion,predicted by the thermal-CFD model, that influences the fatigue life ofthe component. The void space is represented by a unique phase ID inboth the thermal-CFD results and the CA results and is interpreted as avoid by the SCA model. When the framework is used without the CA model,the void space is directly passed to SCA, again through a unique phaseID in the voxel database, and is passed via the CA model when the CAmodel is used. As shown in FIG. 10, panel (b) illustrates these VOIs in3D with the surrounding material removed; panel (c) illustrates thesingle crystal voxel mesh used for FIP prediction; panel (d) showsclustering used for the single crystal, with a void (white) case; panel(e) shows the clusters used for the polycrystal case; and panel (f)shows the clusters used for the polycrystal with void case.

Anisotropic Response of the Polycrystalline VOIs

For the polycrystalline case, where the CA model is employed, theanisotropic stress-strain response at the strain rate of 10⁻⁴s⁻¹ for thefirst three VOIs with loading in the x-, y-, and r-directions is shownin FIG. 11; an anisotropic and location dependent response emerges fromthe crystallographic structure. Each of these VOIs are 40 voxels×40voxels×40 voxels. A voxelated mesh is constructed directly from the VOI,with material properties assigned to each voxel corresponding to theknown properties for the crystal predicted by the CA model. The periodicboundary condition assumption can result in spurious stresslocalizations near the boundary of these volumes. To counteract this,the five surface layers of voxels are neglected for the resultspresented below. This is similar to the relatively common “buffer zone”approximation.

Performance Prediction: Fatigue Crack Incubation Life

Relative values of N_(inc) or FIP can provide insight into the impact aparticular defect, or perhaps a class of defects such as lack of fusionvoids, has upon the expected lifetime of the material under cyclicloading. For example, here we compare the computed FIP value for fourdifferent VOIs. The first two show results using the framework withoutthe CA model, and the second two show results with the CA model. In eachof these sets, one result is taken from Case 1 (which exhibits voids) inTable 1-3 and the other is from Case 2 (where no voids are observed).

The first is a reference VOI: a simple single-crystal cube, representinga defect-free build (without the CA model) The second VOI is the samesingle-crystal cube, this time with a void predicted by the thermal-CFDmodel. This comes from Case 2 in Table 1-3, where larger hatch spacingresults in insufficient fusion. The third VOI incorporates the CA model,making this a polycrystalline simulation. The fourth VOI adds the voidto the poly crystal line volume. The simple, cubic voxel mesh underlyingall four of these VOIs is shown in panel (c) of FIG. 10. The subsequentpanel (d)-(f) of FIG. 10 show the clusters used for SCA, computed fromthe elastic DNS, for the latter three RUCs, respectively.

FIG. 12 shows experimental strain life data (points) for HCF, and fourpredictions of fatigue life: (1) a single crystal without any defects(baseline data). (2) a single crystal with a void predicted with thethermal-CFD model, (3) the full polycrystalline fatigue VOI shown inFIG. 10, and (4) this same growth pattern, including a void. In thesingle crystal condition without a void there arc no stress localizers,and the highest life is expected. The introduction of a void decreasesthe expected life substantially, as local stress concentrations occuraround the void. In the poly crystal line VOI grain boundaries act asstress localizers, and even without a void the fatigue life is reducedfrom the reference VOI; however, the introduction of a void has lessimpact than in the single crystal VOI, since the worst caseconfiguration—a high Schmid factor grain near the void—has not occurred.The extremely high predicted life at very low strain amplitude is likelybecause our current model does not capture the transition of failuremechanism thought to occur between high cycle fatigue and ultra-highcycle fatigue in Ti-6Al-4V. While not necessarily meaningful, it doesnot impact the current demonstration of the framework unduly.

The trends between different cases match qualitatively with experimentalresults demonstrated for SEBM. Subjecting the material to hot isostaticpressing (HIP) is known to suppress voids, whereas heat treatment andas-built material retains any voids introduced during manufacturing.Thus, HIP SEBM results and non-HIP SEBM results are comparable to thepolycrystal no-void and void results respectively shown in FIG. 12: inboth the simulation and experiment reduction of void content results inhigher fatigue life.

This set of examples shows that the framework and models here cancapture, at least qualitatively, the influence of process parameters(hatch spacing in the example shown) on expected material performance.This influence is captured by directly considering the change inmaterial structure, i.e., the introduction of a void cause by the lackof fusion, that results from a change in the processing conditions. Themethod presented is general: the impact of process parameter selectionupon expected material performance can be predicted using thisframework. Further, we have demonstrated that the framework is flexibleenough that models can be added, removed, or exchanged depending on thelevel of detail desired. In this case, we show the framework operatingboth with and without a prediction of grain growth via the CA model.

In sun, we have demonstrated a process-structure-properties-performanceprediction framework for additive manufacturing that connects models foreach aforementioned stage and requires only basic material propertiesand processing conditions. The framework provides a means to capture AMprocess that is extensible and flexible: additional models could beincluded (e.g., post-processing), and different types of analysis couldbe considered (e.g., continuum scale). This implementation containsmicroscale models that incorporate relevant processes and conditions forselective electron beam melting of Ti-6Al-4V. This enables computationaldesign of the SEBM process, at least within the constraints of modelapplicability, using parameters derived from AM machines, raw powder,and studies of wrought material.

Outlook for the Individual Modules:

The largest limitation of the thermal-CFD model currently is the lack ofan explicit gas or vapor phase. This prohibits the prediction of poresdue to entrained gas. Residual stress, an important factor in all AMprocesses, is also not captured. This important facet is the focus ofongoing work.

One limitation of the current grain formation model is the inability topredict solid-state phase transformations during reheating. Thus, thisprediction is thought to be applicable to the final layer of a build;reheating due to subsequent passes of the beam for interior layers isknown to cause grain coarsening and solid state phase transformation.However, the grain shape predicted is that of the β-phase (bcc), but αand α+β is known to be dominant in the final part. Under solid statephase transformations, the β to α transition creates smaller, a grainsfrom the parent β-phase; from this, a material with mixed β+α is usuallyexpected. There is ongoing work to develop a module that captures solidstate phase transformations and grain coarsening effects, but this awork in progress. It would be possible to include such a model in theframework presented here, but the simulations shown do not capture allthe details of a multipass build in which a series of phasetransformations have taken place.

The SCA model, though faster than equivalent FEM models, is stilllimited in the number of grains it can capture by the computationalcomplexity of the CP model. Very large deformation predictions may beinaccurate, since grain boundary motion is not captured and otherrelatively more complex modes are discounted. Furthermore, even thoughthe grain geometry used is that of the phase, mechanical properties ofthe a phase (hcp) are used to more realistically capture mechanicalbehavior. This may be a reasonable approximation, and could easily becorrected if a model for solid state phase transformation is introduced.

Outlook for the Framework:

We claim the framework demonstrated here is sufficiently general toinclude macroscale simulations, if needed, or any other related modulerequired. We show the addition of a CA model as an example of this.However, one could envisage the addition of further hubs. One optionwould be a design hub, which might include modules for CAD/CAE tools,slicing and machine control, and/or topological optimization. Anotherpossible hub might implement models to capture post-processing stepssuch as hot isostatic pressing (HIP), heat and/or chemical treatment,and surface finishing.

Example 2 Implementation and Application of the MultiresolutionContinuum Theory

The multiresolution continuum theory (MCT) is implemented in FEA with abespoke user defined element and materials. In this exemplary study, wefocus on the formulation and implementation of the MCT model. A simpledog-bone model is used to validate the code and study the effect ofmicroscale parameters. The ability of the MCT method to simulate thepropagation of a shear band in simple shear geometry is shown. Thelength scale parameter is demonstrated to influence shear band width.Finally, we present a simulation of serrated chip formation in metalcutting, a case where accurate prediction of shear band formation iscritical. The advantages of the MCT over conventional methods arediscussed. This work helps elucidate the role of the length scale andmicroscale parameters in the MCT, and is a demonstration of a practicalengineering application of the method: the simulation of high speedmetal cutting (HSMC).

Implementation of Multiresolution Continuum Theory

Overview of Multiresolution Continuum Theory

In the multiresolution approach, deformation at a continuum point isdecomposed into the homogeneous deformation and a set of inhomogeneousdeformations; each inhomogeneous measure is associated with acharacteristic scale in the microstructure. This introduces a set ofmicro-stresses in the governing equations that represent a resistance toinhomogeneous deformation. These extra degrees of freedom enable amicrostructure scale resolution to be obtained in the continuum solutionwhile requiring much less computational effort than a detailedmicrostructure analysis. A general multiresolution framework is thusformulated in which:

(1) The material structure and the deformation field are resolved ateach scale of interest.

(2) The resulting internal power is a multi-field expression withcontributions from the average deformation at each scale, i.e., theoverall properties depend on the average deformation at each scale.

(3) The deformation behavior at each scale is found by testing themicromechanical response. Constitutive relations can be developed ateach scale.

An averaging operation is used to make the method consistent withcontinuum mechanics. FIG. 13 gives a schematic for the framework of athree-scale homogenization.

In the multiscale continuum, the kinematics of the body are described bya macro-velocity field, v, and (N−1) micro-velocity-gradient fields,where N is the number of scales. To enable its implementation in finiteelements, we describe the internal (Eq. (2-1)), external (Eq. (2-2)) andkinematic (Eq. (2-3)) virtual power in variational form, as follows:

${\delta \; P_{int}} = {\int_{\Omega}{\left\lbrack {{\sigma \text{:}\delta \; D} + {\sum\limits_{I = 1}^{N - 1}\left\lbrack {{\beta^{I}\text{:}{\delta \left( {D^{I} - D} \right)}} + {{\overset{\_}{\beta}}^{I}\mspace{11mu} \vdots \mspace{11mu} \delta \; D^{I}\nabla}} \right\rbrack}} \right\rbrack d\; \Omega}}$${\delta \; P_{ext}} = {\int_{\Gamma_{I}}{\left\lbrack {{\overset{.}{t}\; \delta \; v} + {\sum\limits_{I = 1}^{N - 1}{T^{I}\text{:}\delta \; D^{I}}}} \right\rbrack d\; \Gamma}}$${\delta \; P_{kin}} = {\int_{\Omega}{\left\lbrack {{\rho \; \overset{.}{v}\; \delta \; v} + {\sum\limits_{I = 1}^{N - 1}{\left( {{\overset{.}{D}}^{I}I^{I}} \right)\text{:}\delta \; D^{I}}}} \right\rbrack d\; \Omega}}$

where the symmetry of the macro-stress, σ, allows us to use the rate ofdeformation, D^(I), rather than the full velocity gradient. Themicrostresses, β^(I), and microstress couples, β ^(I), are defined aswork conjugate to the relative microscale rate of deformation,(δD^(I)−δD), and δL^(I)∇. Here,

$I^{I} = {\frac{1}{\Omega^{I}}{\int_{\Omega_{I}}{{\rho^{I} \otimes d}\; \Omega_{I}}}}$

is defined as the inertial tensor at a microscale I, and ρ=ρ+Σ_(I=1)^(N-1)ρ^(I) is the macroscopic density plus each of the microscaledensities. This description of kinematics discards the higher orderterms and assumes no relative rotation between scales. The tractionforce and traction couple at each microscale are given by t and T¹. Onecould consider σ to be the conventional Cauchy stress tensor, which isaugmented by additional stress terms to define the mechanical responseof much smaller heterogeneities within the material. Another way to saythis is that the macroscale provides high-level homogenization of thestress while the microscale captures inhomogeneous deformation,perturbations about the homogeneous mean, caused by the microstructure.

The governing equations are as follows,

(σ−Σ_(n=1) ^(N)β^(n))Δ+b=ρ{dot over (v)} in Ω  (2-4)

∇·β ^(n)−β^(n) +B ^(n) =γ·I ^(n) in Ω  (2-5)

where b is the body force, B^(n) is the body couple stresses whichbalance the micro-stress, and γ is the micro acceleration. The boundaryconditions are:

t−N·(σ−Σ_(n) ^(N)β^(n))=0 onΓ _(t)  (2-6)

R ^(n) =r ^(n) N=N·β ^(n):(NN) on Γ_(t)  (2-7)

there t is the traction force, R^(n) is the double traction force, r^(n)is the micro-acceleration at each scale. To solve the multiresolutioncontinuum governing equations, the constitutive relationships {dot over(σ)}(D), {dot over (β)}^(n)(D^(n)−D), and {dot over (β)} ^(n)(D^(n)∇)are required.

The MCT can be thought of as a generalization of Fleck and Hutchinson'sstrain gradient theory to an arbitrary number of scales with bothmicropolar and microstretch components. This broadens the applicabilityof the method. Furthermore, the governing equation of the MCT can besolved with a conventional FEM approach. The length scales can berelated to the material microstructure, such as the spacing of voids andinclusions, which give it a more clear meaning than the intrinsic lengthscale of strain gradient theory, and is more easily derived frommicrostructure observation.

Implementation of the MCT

The multiresolution model developed in this study decomposes thedeformation across different microstructural length scales. The macrodeformation is described by conventional degrees of freedom and extradegrees of freedom are introduced to describe the deformation at themicroscale. The macroscale is calculated based on standard continuumtheory. The microscale includes micro-stretch D(1) continuum andmicromorphic continuum D (1)+W(1) components. Therefore, themultiresolution continuum may be generalized to n microscales, and eachscale can be micro-stretch D(n) or micromorphic D(n)+W(n). Each scalerequires a user material.

FIG. 14 shows the relationships of the subroutines used in MCTimplementation, where MCT_userMatMicroscale_*.f andMCT_userMatMacroscale.f are the user materials subroutines for the microand macro domains. The auxiliary function subroutine MCT_auxFunc.fcontains the functions that are used by both subroutines needed for theuser element with the implementation of the MCT, such as Gaussintegration, computation of inertia and the subroutines to calculate thestrain and strain rate. It can be called by the user materialssubroutines to finish the update of the stress and strain. Thesubroutine MCT_forceInt.f is used to calculate the internal force bycalling the user materials subroutines and the auxiliary functionsubroutine. The parameter subroutine MCT_parameters.f gives the inputparameters for user element, such as number of micro scales, number ofGauss points and number of history variables. The MCT_mass.f containsthe subroutine used to compute the mass matrix for the user element.These three subroutines are called by the user element subroutine foreach solve.

Macroscale Material Model

A modified version of the classic homogenization-based Gurson (GTN)model is employed at the macroscale. An additive decomposition of therate of deformation tensor, D, into elastic and plastic parts allows forthe description of an objective rate of Cauchy stress. The material flowrule is:

$\begin{matrix}{{\Phi \left( {\sigma,Q} \right)} = {\left( \frac{\sigma_{eq}}{\sigma_{y}} \right)^{2} + {2q_{1}{\cosh \left( \frac{2q_{3}\sigma_{m}}{2\; \sigma_{y}} \right)}} - \left( {1 + {q_{2}f^{2}}} \right)}} & \left( {2\text{-}8} \right)\end{matrix}$

where σ_(eq), σ_(y), and σ_(m) are the equivalent, yield, andhydrostatic stresses for the matrix, q₁, q₂, q₃ are material constantsadded to the original Gurson model to improve the prediction result, andf represents the current void volume fraction. The evolution of voidvolume fraction is taken as the sum of void growth and void coalescencerates, such that

$\begin{matrix}{\overset{.}{f} = {{\overset{.}{f}}_{g} + {\overset{.}{f}}_{c}}} & \left( {2\text{-}9} \right) \\{{with},} & \; \\{{\overset{.}{f}}_{g} = {\left( {1 - f} \right){{tr}\left( {\overset{.}{D}}_{p} \right)}}} & \left( {2\text{-}10} \right) \\{{\overset{.}{f}}_{c} = {k_{w}f\; {\omega (\sigma)}\frac{S\text{:}\mspace{14mu} D^{p}}{\sigma_{e}}}} & \left( {2\text{-}11} \right)\end{matrix}$

Where {dot over (f)}_(g) is the void growth rate, {dot over (D)}_(p) isthe rate of plastic deformation, {dot over (f)}_(c) is the rate of voidcoalescence, and A is the acceleration factor. A power law hardeningrelationship is used, given by

$\begin{matrix}{\overset{\_}{\sigma} = {{\sigma_{y\; 0}\left( {1 + \frac{\overset{\_}{\epsilon}}{\epsilon_{0}}} \right)}^{n}\left( {1 + \frac{b\left( {1 - {\exp \left\lbrack {c\left( {T - T_{r}} \right)} \right\rbrack}} \right)}{\exp ({cT})}} \right)}} & \left( {2\text{-}12} \right)\end{matrix}$

where σ is the flow stress, σ_(y0) is the initial yield stress, ϵ is theequivalent plastic strain, ϵ₀ is the reference strain, T is thetemperature and can be calculated by

${T = \frac{0.9\; \overset{\_}{\sigma}\mspace{11mu} \overset{\_}{\epsilon}}{\rho \; C_{p}}},$

ρ is material density and C_(p) is the specific heat, T_(r) is the roomtemperature, n, b and c are material constants.

An iterative Newton-Raphson scheme is used to solve the resultingequations at each step. The void volume fraction f, effective plasticstrain rate {dot over (ϵ)}, and matrix flow stress σ_(y) are all updatedbased on the new D. The homogenized void growth behavior is taken torepresent the response of the primary population of voids, includingthose initiated at inclusions, as the material is deformed. As thematerial damages, its ability to resist load degenerates and eventuallyvanishes. This is captured by using a modified the void volumeparameter, f*, that causes damage to accelerate after a critical voidvolume fraction called the coalescence point, f^(crit) has been reached.At f^(ult), the material is considered to have lost its load carryingcapacity, and that element is removed. Specifically, this is defined by

$\begin{matrix}{f^{*} = \left\{ \begin{matrix}f & {for} & {f < f^{crit}} \\{f^{crit} + {\frac{\partial f}{\partial f^{*}}\left( {f - f^{crit}} \right)}} & {for} & {f^{crit} < f < f^{ult}} \\f^{ult} & {for} & {f > f^{ult}}\end{matrix} \right.} & \left( {2\text{-}13} \right) \\{\frac{\partial f}{\partial f^{*}} = \left\{ \begin{matrix}1 & {{{for}\mspace{14mu} f} < f^{crit}} \\\frac{q_{1}^{- 1} - f^{crit}}{f^{ult} - f^{crit}} & {{{for}\mspace{14mu} f^{crit}} < f < f^{ult}} \\0 & {{for}\mspace{14mu} > f^{ult}}\end{matrix} \right.} & \left( {2\text{-}14} \right)\end{matrix}$

Microscale Material Model

At the microscale, the microstress β and microstress couple β are usedto construct the plastic potential:

Φ=σ_(e) ^(H)−σ_(Y)(E ^(p))  (2-15)

With σ_(e) ^(H) given by

$\begin{matrix}{\sigma_{e}^{H} = \sqrt{{\frac{3}{2}\beta^{I,{dev}}};{\beta^{I,{dev}} + {\left( \frac{\sqrt{18}}{l^{I}} \right)^{2}{\overset{\_}{\beta}}^{I,{dev}}\text{:}{\overset{\_}{\beta}}^{I,{dev}}}}}} & \left( {2\text{-}16} \right)\end{matrix}$

where I indicates the I^(th) concurrent microscale, and the deviatoricpart of the microstress and microstress couple are given by

$\begin{matrix}{\beta^{{dev}\nabla} = {2G^{I}{\overset{\_}{D}}^{Ie}}} & \left( {2\text{-}17} \right) \\{{\overset{\_}{\beta}}^{{dev}\nabla} = {\frac{l^{I}}{12}2{G^{I}\left( D^{I} \right)}^{e}}} & \left( {2\text{-}18} \right)\end{matrix}$

where D^(I) is easiest defined in indicial notation as D_(ijk)^(I)=½(L_(ij,k)I+L_(ij,k) ^(I)). The deviatoric strain rate, {tilde over(D)}′={tilde over (D)}−⅓ tr({tilde over (D)}), is defined in terms ofthe relative deformation between the micro and macro scales, given by:{tilde over (D)}=D^(I)−D. In this work, the superscript I is neglected,as only one micro is used.

The hardening relationship is given by

σ ⁽¹⁾=σ_(y0) ⁽¹⁾(1+dϵ ⁽¹⁾)  (2-19)

where σ ⁽¹⁾ is the flow stress for scale (1), σ_(y0) ⁽¹⁾ is the initialyield stress, ϵ ⁽¹⁾ is the equivalent plastic strain, and d is amaterial constant.

Validation of Code and Study of Effect of Microscale Parameters

Validation

If these equations are implemented correctly, the solution should reduceto that of a standard element when the microscale is turned off. Wefirst do this comparison to validate our code. First, one microscale isused for the dog-bone model shown in FIG. 15. The geometry is 36 mm×24mm×4 mm, and contains 22840 elements. The bottom surface of the dog-boneis fixed, and the top is given a fixed displacement.

For simplicity of these tests, we set q₁=q₂=q₃=0 in the macroscalematerial laws, so it reduces to

ϕ=σ ²−σ_(eq) ²=0  (2-20)

This is a standard J2 yield criterion. Quasi-static state and dynamiccompression mechanical properties experiments on hardened AISI 1045steel (45HRC) are performed at a range of temperature from 20 to 800° C.and strain rate from 10⁻³ to 10⁴/s by electronic universal testingmachine and Split Hopkinson Pressure Bar. From which the stress-straincurves are obtained. The material parameters used in macroscale can becalibrated by fitting the curves of stress and strain and are listed inTable 2-1. The materials parameters used in microscale are determined,the length scale is ascertained by considering the controllingmicrostructural space, in this case the void spacing, as listed in Table2-2.

TABLE 2-1 Material constants for J2 used in the macroscale law forhardened AISI 1045 (HRC45) ρ σ_(y0) C_(p) E (GPa) μ (kg/m³) (GPa) n b cε₀ (J/kg · K) 200 0.3 7760 0.7827 0.14 0 0.00793 0.03 465

TABLE 2-2 Material constants used in the microscale law for AISI 1045(HRC45) E (GPa) μ σ_(y0) (GPa) l (m) 20 0.3 0.078 0.02e−3The comparison of the simulation results for a standard element and theuser defined element is shown in FIG. 16. The stress-strain curves arethe same. This indicates that the macroscale portion of the codeproduces the expected results. When MCT is used, different materialbehavior is exhibited, as shown in FIG. 17. The microscale tends toresist inhomogeneous deformation, therefore a larger stress at the samestrain is measured when the microscale is enabled.

Effect of Microscale Parameters

The microscale parameters influence the macro strain-stress curves.FIGS. 18-22 show the effect of microscale parameters on the macrobehavior under different conditions. The basic microscale parameters arethe Young's modulus E⁽¹⁾ and length scale l. The influence of these twoparameters is shown for five different sets of the remaining microscaleparameters. FIG. 18 shows these effects when a purely elastic microscalematerial is used. The trend is that the equivalent stress increases withincreasing E⁽¹⁾ or l. When plastic behavior for the micro scale is used,the effect of the microscale parameters depends on the hardening of themicro scale, as shown in FIGS. 19 and 20. If microscale hardening isused, the effect of Young's modulus E⁽¹⁾ and length scale l are reduced.For a perfectly plastic microscale material, the Young's modulus doesnot influence the overall stress-strain curve, and the impact of l issubstantially reduced. Generalizing, when the slope of the stress-straincurve during plasticity for the microscale is smaller, the influence ofYoung's modulus is also smaller.

A thermally softening material (by setting b=0.15) may be used at themacroscale with similar results: the influence of length scale andYoung's Modulus at the microscale varies between relatively large for apurely elastic material to relatively small for a perfectly plasticmaterial, just as for the cases shown above.

Parallel Efficiency

A simple scaling study addressed the ability of a FEA code to operate ina parallel environment, and the impact that our user element formulationhas upon that ability. This study was done using an illustrativedog-bone geometry, this time with 5250 elements and 20124 nodes.

FIG. 23 shows the efficiency (left) and speedup (right) observed understrong scaling for three cases: a conventional 3D brick element withplasticity, the MCT element with only the macro scale, and the MCTelement with one microscale. For reference, the total time for each ofthese simulations with one processor is 498 s, 30536 s, and 47589 s,respectively. The latter case demonstrates the impact of the addedcomputations required to solve the material law at the microscale, whilethe former two cases demonstrate the impact of using the MCT element,absent of the additional solves required for the microscale, and astandard element on the same mesh. Note that because MCT is a multiscalesolution, it is much slower on the same mesh but contain moreinformation. These results seem to indicate that the added element-levelwork required by a custom user element decreases the relativecommunication penalties, allowing the MCT implementation to remainefficient even with a relatively low element count to processor countratio, e.g., 82 elements/processor for the final 64 processor case. Thisstudy was performed on a 176 core, distributed memory cluster computerwith eight cores/node. This means that, while MCT is slower in absoluteterms than a conventional homogenizing plasticity law (and much fasterthan a direct numerical simulation of the microstructure), moreprocessors may be used with relatively high efficiency to solve the sameproblem.

Simple Shear

MCT was applied to simulate shear band formation under simple shear, andthe change of shear band width with mesh size and length scale wasstudied. The GTN model is used as the macroscale law with the materialparameters shown in Table 2-3. The materials parameters used in themicroscale are the same as those used in the dogbone model. An arbitrarylength scale is used to demonstrate the effect of the length scale inthe MCT model.

TABLE 2-3 Partly material constants for the GTN model used as themacroscale law for AISI 1045 (HRC45) q₁ q₂ q₃ f_(crit) f_(ult) 1.5 1.02.25 0.15 0.37

To simulate a simple shear loading scenario, we use the model shown inFIG. 24. It has one element wide (10 μm) pre-crack. A constant velocityof 50 mm/s is instantaneously applied to the top surface of the domainin the x-direction. The top surface is constrained in the y-directionand the bottom surface is constrained in both the x- and y-directions.The continuum result with conventional FEA is shown in FIG. 25, wherethe shear band localized in one layer of elements. When a finer mesh isused, a similar shear band is observed to localize within one layer ofelements, demonstrating that shear band formation is mesh dependent.Here elements with equivalent strain larger than 0.5 are considered tobe in the shear band.

When the MCT is used, shear bands with consistent width, independent ofmesh size, are predicted (panel (a) of FIG. 26. With a decrease oflength scale, the width of shear band decrease (panels (b)-(c) of FIG.26. In order to analysis the relationship between the shear band widthand length scale, the width of shear band is defined as the distancebetween constant-strain fringes where the local total effective plasticstrain is equal to 0.5. It can be given by the strain fringe pattern. Weobserve a nonlinear scaling between the width of the shear band and thelength scale, as demonstrated in Table 2-4.

TABLE 2-4 Influence of length scale on shear band width Length scale(μm) Shear band width (μm) 100 80 40 60 20 40

When two microscales are activated in MCT for the model shown in FIG.24, the difference in shear band formation is minor compared to theresults from one microscale, as seen in panels (a)-(c) of FIG. 28compares the strain distribution along the direction perpendicular tothe shear zone. The width of the shear band is little larger, and thestrain distribution is more uniform for two microscales than for onemicroscale.

The above comparison between one microscale and two microscales resultsshows that the MCT model with one microscale is sufficient to capturelocalization within the model. However, the two microscale result issmoother than the result of one microscale, as shown in FIG. 28. Thus,by applying different levels of length scale the shear band formationcan be smoothed to different extents. When unrealistic shear bandformation is observed compared to experiments, it may be possible todeploy different number of length scales to obtain a realistic shearband formation.

Application of MCT to Metal Cutting

An example study of metal cutting using MCT is presented in thissection. A 3D formulation with boundary conditions to approximate planestrain through the depth of the cut is used. This quasi-2D model hasfour primary components, as presented in FIG. 29. These are the cuttingtool, an eroding layer, and the chip and base sections of the materialto be cut. The selection of this comparison of our results toexperimental and simulation results of other authors. We aim to capturethe saw-tooth chip formation characteristic of shear localization duringhigh speed metal cutting.

The tool here is assumed to be stiff compared to the workpiece and istreated as a rigid body with nominal Young's Modulus of 2.0×10⁴ GPa andPoisson's Ratio of 0.25. The tool has a rake angle of 20° and is sharplyhoned. Cutting speed, defined by the velocity of the tool with respectto the stationary workpiece, is initially set to 105 m/min.

TABLE 2-5 Material constants used in the macroscale GTN law for AISI316L Steel G (GPa) K(GPa) ρ (kg/m³) σ_(y0) (GPa) n q₁ q₂ q₃ f^(crit)F^(init) A 29.86 49.77 7780 0.980 0.25 0.91 1.9 0.95 0.1 0.001 3

TABLE 2-6 Material constants used in the microscale J2-based law forAISI 316L Steel G (GPa) K (GPa) ρ (kg/m³) σ_(y0) (GPa) L (m) 0.08960.149 70839 0.098 0.035e−3

The material laws employed within the three sections of the workpiece(chip, eroding layer, and base) are identical (defined in Tables 2-5 and2-6); the distinction between sections is made for clarity of purpose.Currently, microscale parameters are defined as 0.003 of the macroscaleparameters. The eroding layer uses an eroding contact law to achieveelement deletion based on critical effective plastic strain of 0.5,facilitating the material removal process with a predeterminedseparation location. Elsewhere, we chose to model contact with asegment-based soft constrain algorithm for both surface and edge-to-edgepenetration. While the particulars of the contact laws do substantiallyimpact the solution (including cutting force and chip morphology), thesedetails are omitted here to retain a focus on the MCT aspects of thispaper. Static and dynamic coefficients of friction between the tool andworkpiece are 0.28 and 0.25 at 105 m/min to represent dry cutting. Thecontact utilizes node to surface contact between tool and workpiece,where workpiece is considered as a node set to reduce possiblepenetration during the cutting process. Similarly, static and dynamiccoefficients of friction are 0.27 and 0.25 at 150 m/min and 0.25 and0.23 at 210 m/min. Geometrically, the chip is initially slightlyreclined. Our preliminary work indicates that this does not affect thefully evolved chip formation, and it eases the simulation process byreducing initial mesh distortion. The initial geometry uses 0.2 mm chipthickness. The mesh is constructed with 8 node brick elementsimplementing MCT with a user-defined element. A single layer of elementsis used in the depth direction, with plane strain boundary conditions(zero through-plane displacement) are applied. This mimics cutting andturning operations where the through-plane thickness is generally largeand the workpiece experiences plane strain. The initial element edgelength is 0.01 mm.

TABLE 2-7 Predicted cutting forces normalized to unit thickness andsurface residual stress, with reference experimental data for comparison(Cutting depth at 0.2 mm and Speed at 105 m/min) Experimental SimulationCutting Force (N) 250 130 Surface Residual Stress (MPa) 600 373

The results from the cutting geometry are shown in FIG. 30, after 0.2ms. Panel (a) of FIG. 30 shows contours of J2 stress at 0.2 ms; note thecharacteristic saw-tooth chip with stress localized about the shearbands. In panel (b) of FIG. 30, the formed shear is comparable toexperimental shown. Contours of J₂, units of stress squared, are plottedwithin the workpiece. This is related to the von Mises equivalent stressby:

σ_(Mises)=√{square root over (3·J ₂)}  (2-21)

The contours observed here match qualitatively with previous steelcutting simulations. Maximum stress occurs in the primary shear zone.Limitations, i.e., geometry, of the model are such that only the surfacelayer residual stress can be predicted.

As noted, one of the benefits of the MCT method, and more generallygradient-based methods, is that the shear band behavior is generallyrelatively mesh independent. The localization is controlled by thelength scale introduced rather than the mesh. It is convenient to plotcontours of effective plastic strain to highlight the localizationeffects, as has been done by in panel (b) of FIG. 30. This shows thatwhile still predominantly within one element, the deformation has beendistributed to nearby elements as well, with elevated levels of plasticstrain three elements wide. Thus, the measured shear band width is 0.028mm, close to the 0.035 mm width measured from the experimental results.

The XY components of microstress and macrostress are plotted as contoursat 0.22 ms shown in FIG. 31. Clearly, the microscale is in effect duringcutting process and contours show that the microstress is on the samemagnitude as macrostress. The microstress is mainly distributed aroundshear band, which is consistent with the role described earlier.

The surface residual stress is 299 MPa at 0.1 mm cutting depth, where310 MPa is predicted. This lower value can be attributed to the 15° rakeangled used versus 20° of our model. This follows the trend identifiedexperimentally.

The cyclic cutting force we observe is related to the serrated chipformation: force drops when a shear instability event occurs andincreases again until sufficient stress to cause a localization is againreached. This is also shown by the four states in FIG. 32. Each insetshows the progress of the cut at the point indicated by the dot alongthe force-time curve at the bottom. This collection shows the formationof the first (panel (α) of FIG. 32, 0.08 ms), second (panels (b)-(c) ofFIG. 32, 0.12 ms and 0.14 ms), and third (panel (d) of FIG. 32, 0.16 ms)shear bands. Clearly, formation of shear band is associated with forcedrop. A simple force normalization is used to scale the cutting force toa unit-thickness equivalent. The average cutting forces, collected inTable 2-7, is less than experimentally measured forces. Reported cuttingforce is between 250 N/mm to 400 N/mm and residual stress is 600 MPa.Different element sizes were tested, and have negligible effect on thecutting force, partly due to the regularization from the material lengthscale. The difference between simulation results and experimentalresults may be accounted for by considering the simplifications of themodel, e.g., a sharp tool, 20° rake angle, a thin uncut region, and aquasi-2D plane strain representation, and expected model andexperimental uncertainty.

Nonetheless, the MCT model has been applied to model metal cutting withmodest success. Comparable shear band formation and surface residualstress are observed through simulation. Cyclic cutting forces areobserved, which agrees with experimental results.

The MCT shown here requires calibration for all the material parametersat all scales (e.g., microscale 1, microscale 2) for each specificmaterial used in the computation. This can be burdensome, particularlybecause of the microscales which, while motivated by physicallyhierarchical material structure, in practice simply regularize thematerial behavior. Such calibration might be replaced with a data-drivenmachine learning approach, to improve the applicability of MCT ondifferent material system. In addition, the lower scale response,instead of using material laws, could be directly linked with arepresentative volume element (RVE) computation of the designatedmicrostructure. Further developments of the MCT model could use an RVEhomogenization to provide both constitute response and real-timemicrostructure evolution at the microscale. We suspect that furtheroptimization the contact parameters and formulation used for the cuttingsimulation could produce more physically accurate results.

In sum, our MCT implementation was validated with a simple dog-bonemodel. The effects of microscale parameters were discussed. Thepropagation of shear band in simple shear was simulated to show thechange of shear band width with length scale and investigate the meshrelevance. The simulation of the shear band of serrated chip in metalcutting was also been done to show the advantages of the MCT whencompared with conventional continuum theory. According to this study,the MCT can be implemented in a FEM solver as a user element subroutine;the influence of microscale parameters on the macro strain-stress curvedepends on the hardening law used in micro zone. For both materials withand without thermal softening, the influence of microscale parametersbecomes more obviously when the microscale material response hardensmore; from a simulation of simple shear, the shear band width was shownto depend on the length scale parameter, not the mesh size, when usingthe MCT approach; and because of its ability to capture shear bandswell, the MCT was used to model shear band formation during metalcutting, showing reasonable agreement with published experimentalresults.

Example 3 Data-Driven Mechanistic Modeling of Microstructural Influenceon High Cycle Fatigue Life of Nickel Titanium

In artificial heart valve frames and arterial stents made from shapememory alloys such as Nickel Titanium (NiTi), failure can occur in highcycle fatigue (HCF). Mechanical failure of these devices is thought tohave negative clinical implications. It is generally accepted that, froma materials perspective, failure is driven by extrinsic defectsintroduced during material processing, specifically oxide inclusions.

In this exemplary study, we focus on a class of defects knowncollectively as stringers, which result from the breakup of oxide orcarbide particles during the wire-drawing process conventionally used tofabricate these biomedical devices. Specifically, a data-drivenmechanistic modeling technique is applied to a system representative ofa broken-up inclusion (stringer) within drawn Nickel-Titanium wire ortube, e.g., as used for arterial stents. The approach uses adecomposition of the problem into a training stage and a predictionstage. It is used to compute the fatigue crack incubation life of amicrostructure of interest under high cycle fatigue. A parametric studyof a matrix-inclusion-void microstructure is conducted. The resultsindicate that, within the range studied, a larger void between halves ofthe inclusion increases fatigue life, while larger inclusion diameterreduces fatigue life.

Methods: Data-Driven Mechanistic Modeling

Crystal Plasticity Self-consistent Clustering Analysis (CPSCA) is anextension of the existing Self-consistent Clustering Analysis (SCA)approach to (poly)crystalline metals achieved by using a crystalplasticity (CP) material law. An overview of the SCA method is shown inFIG. 33: from left to right the figure shows (1) a simplifiedrepresentation of the material system (loosely called an RVE), with halfthe blue matrix material hidden showing the gray inclusion and red void;(2) contours of the xx component of strain in the matrix from a linearelastic analysis; (3) the clusters computed from the strain solutionusing k-means clustering; (4) the plastic shear strain field from CP,computed in the prediction stage; and (5) the FIP.

The SCA approach defines a database of training data called the DirectNumerical Solution (DNS); construction of this database is madetractable by computing only the elastic response of the RVE using an FFTor FEM solver. The database is used to redefine the number of straindegrees of freedom (DOFs). In the prediction stage, the RVE response isthen solved using a self-consistent solution to the Lippmann-Schwingerequation with any constitutive law, in this case a crystal plasticitymodel. Since the number of DOFs has been reduced, complex load paths(e.g., cyclic loading) and expensive constitutive laws can be usedduring the prediction stage with relatively little impact on totalcomputational time.

Derivation

This method is derived from the first order homogenization of a boundaryvalue problem within an RVE. The local strain field ϵ^(m) is defined asthe sum of a macroscopic, homogeneous strain field ϵ and a fluctuatingterm, i.e., ϵ^(m)=ϵ+ϵ*. To satisfy the Hill-Mandel condition, we assumeperiodic displacement field and anti-periodic boundary traction. Thelocal stress must satisfy equilibrium, that is:

∇·σ^(m)(x)=0, x∈Ω,  (3-1)

where σ^(m)(x) is the local stress tensor at point x in the RVE domainΩ. The local stress σ^(m) can be computed using any constitutiveequation.

Eq. (3-1) is equivalent to the Lippmann-Schwinger equation for firstorder homogenization, as given by:

ϵ^(m)(x)=−∫_(Ω)Φ⁰(x,x′):(σ^(m)(x′)−C ⁰: ϵ^(m)(x′))dΩ(x′)+ϵ, x∈Ω  (3-2)

where C⁰ is a fourth order reference stiffness tensor, and Φ⁰ is theperiodic fourth-order Green's operator.

Training Data

In one embodiment, Eq. (3-2) is solved on a set of arbitrary sub-domainswithin the RVE, rather than using the DNS discretization. With carefulselection of these sub-domains, or clusters, a dramatic reduction in thenumber of strain DOFs is possible with little loss of accuracy.

To make this selection well, clustering is based on a training dataset.Here, we choose to use the strain concentration tensor, A^(m) defined byε^(m)(x)=A^(m)(x):ε, x∈Ω; how this is computed is determined by theloading conditions used during the prediction stage. For each phase inthe material, voxels with similar A^(m) are assigned to one of apredetermined number of clusters I=1 . . . k using k-means clustering.

Predicting Response

To use clusters in place of voxels, solution variables are assumedconstant across each cluster Ω_(I), or (ϵ_(i) ^(m))_(1 . . . k); (σ₁^(m))_(1 . . . k) and likewise for any other local variable. Using thisclusterwise-constant approximation, Eq. (3-2) is rewritten indiscretized form as:

ϵ_(I) ^(m)=−Σ_(J=1 . . . k) D _(IJ) ⁰:(σ_(J) ^(m) −C ⁰:ϵ_(J) ^(m))+ϵ,l=1. . . k  (3-3)

where D_(IJ) ⁰ is the interaction tensor between clusters I and Jdefined by D_(IJ) ⁰=c¹D_(IJ) ¹+c²D_(IJ) ². Numerical variables c¹ and c²are computed and updated by a physically-based self-consistent schemewhich ensures accuracy at each load step. For the definition of D_(IJ)⁰. Clustering, D_(IJ) ¹, and D_(IJ) ² can be precomputed and stored forfuture use. Given a macroscopic, or external, strain ϵ, Eq. (3-3) can besolved using, e.g., Newton-Raphson iterations for the unknowns ϵ₁ . . .ϵ_(k) using these precomputed values.

Constitutive Laws

In the training stage, only isotropic, linear elastic materials areused. Any constitutive law might be used, but a simple one is sufficientfor the training stage, which is only used to generate clusters.

During the prediction stage, the CP formulation of McGinty is solved. Amultiplicative decomposition of the deformation gradient F^(m) intoelastic and inelastic parts is taken. The deformation gradient isapproximated here by I+ϵ^(m). The plastic velocity gradient in theintermediate configuration, {tilde over (L)}P, defined in Eq. (3-4) isused to compute the plastic part of the deformation gradient. The sum ofthe plastic shear velocity of each slip system of a crystal gives thetotal plastic velocity:

{tilde over (L)} ^(p)=Σ_(α=1) ^(N) ^(slip) {dot over (γ)}^((α))({tildeover (s)} ^((α))⊗ñ^((α)))  (3-4)

where ⊗ is the dyadic product, a is a slip system, N_(slip) is thenumber of slip systems, {dot over (γ)}^((α)) is the microscale shearrate, s^((α)) is the slip direction, and n^((α)) is the slip planenormal. A phenomenological power law determines the shear rate for eachslip system.

Fatigue Indicating Parameter

An estimate of the fatigue crack incubation life is computed using afatigue indicating parameter (FIP) suitable for microstructuralanalysis. This was first derived by Goh and McDowell from theFatemi-Socie critical plane approach, defined by:

$\begin{matrix}{{FIP} = {\frac{{\Delta\gamma}_{\max}^{p}}{2}\left( {1 + {\kappa \; \frac{\sigma_{n}^{\max}}{\sigma_{y}}}} \right)}} & \left( {3\text{-}5} \right)\end{matrix}$

where Δγ_(max) ^(p) is the maximum of the cycle-to-cycle change of theplastic shear strain between two consecutive cycles, σ_(n) ^(max) is thepeak stress normal to the plane on which Δγ_(max) ^(p) occurs, σ_(y) isthe yield stress, and κ is a normal stress factor assumed to be 0.55.The computed change in plastic shear strain saturates after just a fewload cycles, giving a stabilized value of FIP.

The FIP defined by Eq. (3-5) can be quite sensitive to the fineness ofthe discretization. Maximum values of the plastic shear strain arereached in regions where the geometry is singular, and a very finediscretization is needed to compute an accurate FIP. To avoid thecomputational cost that such discretization would imply, previousstudies have used on nonlocal averaging to regularize FIP predictions.We average the FIP value computed in Eq. (3-5) on a finite cubic volumeof fixed size that sets the length scale; this cube contains the peakFIP value. We assume the size is related to the grain size in thematrix, which remains constant. The maximum, saturated nonlocal FIP,NFIP_(max), is correlated to fatigue crack incubation life, N_(inc),with

NFIP_(max)=γ _(f)(2N _(inc))^(c)  (3-6)

where γ _(f) and c, multiplicative and exponential Coffin-Manson-likefactors, are used to fit experimental high cycle fatigue data of NiTi.We use the parameters for NiTi determined by γ _(f)=0.0050 andc=−0.8195.

Representative Volume Element for NiTi Inclusions

We represent a stringer with a simplified geometry, which isparametrically varied Modeling assumptions are made to approximate aworst-case instantiation of this geometry. A 3D rendering of an examplegeometry is shown in on the left of FIG. 33. and the 2D slice on theright shows the parameters under consideration. The inclusion phase(gray) is modeled as linear elastic with ten times the Young's modulusof the matrix phase; this is meant to represent a hard, brittleinclusion though the exact choice of Young's modulus is arbitrary. Thevoid has zero stiffness.

The constitutive behavior of the matrix material is assumed to be thatof the austenitic B2 phase of NiTi. B2 has lower yield strength than theB19′ martensite phase, and thus is likely to fail first. Evidencesuggests that during in vivo fatigue, considering only the B2 phase issufficient to capture minimum fatigue life. The single crystal matrix isoriented with Euler angles Φ=90, θ=45, and ϕ=0: the weakest orientationwith respect to the load applied consistent with a worst-caseassumption.

Loading is applied in the axial direction, because most stents designsensure that stringers align with the direction of load. Thus, an averagestrain is applied in the x-direction in FIG. 33. Zero average stress forall components is applied in the other directions. In the trainingstage, A^(m) is computed from monotonic loading at 1.0% strain with afully elastic material law There are 64 clusters in the matrix, and 16each in the void and inclusion phases. During the prediction stage, fourcycles of fully reversed (R=ϵmin/ϵ _(max)=−1) loading is applied usingthe CP material law. A cyclic strain level of 1.0% is used,approximately the high-cycle fatigue life limit because a stent might bedesigned to operate near this limit. FIP is determined from the changein plastic shear strain between cycles three and four, at the point ofpeak stress in the load history.

We consider RVEs with a range of inclusion diameters and void widths.For each case, a voxel mesh with size 281×191×191 was generatedsystematically. The voxel edge length is 0.305 μm. Cases ranged fromvoid width of about 2.74 μm to 21.0 μm, in increments of 1.22 μm;simultaneously, the inclusion diameter was varied between 9.45 μm and25.3 μm with the same increment. The largest diameter corresponds withthe largest inclusion resulting from standard VAR processing of NiTiwire. Void sizes are less commonly reported, though these are roughlyconsistent with experimental observations of our collaborators. Thisresulted in 224 cases for which SCA was run automatically. A cubicaveraging region with an edge length of 12.2 μm was used to compute thenonlocal FIP.

Results

The predicted fatigue lives are shown in FIG. 34, a 3D rendering of theRVE microstructure, with the matrix material hidden and a 2D sliceshowing the parameters void width d and inclusion diameter D. Thegeneral trend is a nonlinear dependence of fatigue incubation life oninclusion diameter, with larger inclusions resulting in lower estimatedlifetimes. The converse effect is seen with increasing void width: awider void tends to increase the fatigue life. This implies that largervoids may not necessarily be more dangerous. A void with small widthbetween hard inclusions seems to act as a crack tip; with increasingwidth, the stress concentration caused by this crack-like featuredecreases, causing fatigue incubation life to increase. Based on acomparison of mesh sizes, the minor oscillations in the results arelikely a result of the discretization of the problem.

Individual results indicate that the highest local FIP lies at theinterface between void, inclusion, and matrix as expected. This isconsistent with previous experimental and computational studies. Thiscan be seen in the final pane of FIG. 33, where this region is coloredred. FIG. 35 shows a response surface of fatigue crack incubation lifeto void width and inclusion diameter, where the estimated number ofcrack incubation cycles decreases with increasing inclusion diameter butincreases with increasing void width between inclusions.

Each RVE has about 10.3 million voxels and each simulation requiredabout 18 CPU-hours to complete. Computing the elastic FFT solution tookabout 0.6 CPU-hours T_(FFT), determining the clustering and interactiontensor took about 17 CPU-hours (in one case, 3.2 CPU-hours (T_(cl)) and13.8 (T_(inter)) respectively), and the prediction stage took about 85seconds (T_(pred)), i.e.,Time=T_(FFT)+T_(cl)+T_(inter)+T_(pred)=0.6+3.2+13.8+0.024≅18. A simplerFEM simulation of an ellipsoidal elastic inclusion in a CP matrix, withabout 20,000 elements, required about 49 CPU-hours. While only onestrain amplitude is shown here, computing the FIP for these samemicrostructures at different strain amplitudes would require re-runningonly the prediction stage; strain-life plots could be constructed veryquickly.

These results show promise for the potential of this method to generatemany simulation results relatively quickly, and thus thoroughly exploreparameter spaces in ways not possible with computationally expensive FEMor FFT approaches. However, there are still several opportunities forimprovement. A finite strain formulation of CPSCA should be investigatedto improve the integration of the CP material law, with a thoroughconvergence analysis with increasing number of clusters. For a morefundamental understanding of high cycle fatigue life of NiTi, a moreaccurate representation of stringers could be considered by modelingstringers formation during forming or using direct representation fromimages. These could be in 3D from x-ray micro-CT, serial sectioning, oran axi-symmetric assumption for 2D sections. Other areas of future workare to capture R>0 conditions that occur in vivo and the initial crimpimposed during device delivery, as well as stringer to load-axismisalignment effects. The first and last might be simple cases to run,but capturing crimp would require changes to the CP formulation.

In this example, we have presented a data-driven, mechanistically-basedmodeling method for predicting HCF crack incubation life. We appliedthis method to predict the fatigue response of a system including a pairof hemispherical inclusions connected by a cylindrical void within asingle crystal matrix modeled with crystal plasticity. We have shown theresults of a parametric study that probes the impact on fatigue life ofinclusion diameter versus spacing, for this simplified geometry. Theinclusion diameter is directly proportional to the estimated fatiguelife, while the void width between inclusions was inverselyproportional.

Example 4 Microstructural Material Database for Self-ConsistentClustering Analysis of Elastoplastic Strain Softening Materials

Multiscale modeling of heterogeneous material undergoing strainsoftening poses computational challenges for localization of themicrostructure, material instability in the macrostructure, and thecomputational requirement for accurate and efficient concurrentcalculation. The self-consistent clustering analysis (SCA) provides aneffective way of developing a microstructural database based on aclustering algorithm and the Lippmann-Schwinger integral equation, whichenables an efficient and accurate prediction of nonlinear materialresponse. The SCA is further generalized to consider complex loadingpaths through the projection of the effective stiffness tensor. It hasbeen shown recently that the SCA can greatly reduce computationalexpense through the use of novel data mining techniques based onclustering. This method uses RVE clustering techniques (see FIG. 37) tocreates a microstructural database from the high-fidelity simulationdata in an offline training stage and homogenizes the reduced systembased on a self-consistent scheme in the online predicting stage. Withonly linear elastic simulations in the offline stage, the SCA hasdemonstrated a powerful trade-off between accuracy and efficiency inpredicting elasto-plastic behavior with no strain softening. The SCA hasalso been shown to accelerate material behavior predictions (e.g.,toughness) within a computational data-driven framework of materialdesign under uncertainty.

In this example, a novel micro-damage algorithm is disclosed thatalleviates the material instability at the microstructural level. Thehomogenized behavior in the strain softening region is not sensitive tothe RVE size. Also, a non-local macroscopic damage model is introducedwhich couples the homogenized stress from the microstructural RVE with amacroscopic characteristic length and a weighting function. This methodalleviates the material instability in the macroscopic model due tomaterial damage. The proposed concurrent simulation framework allows thecomputation of the macroscopic response to explicitly consider thebehavior of the separate constituents (material phases), as well as thecomplex microstructural morphology.

Multiscale Damage Model with a Micro-Damage Algorithm

Mechanics

Material fracture is sensitive to the microstructure and evaluatingthese effects requires a multiscale modeling approach which incorporatesthese details. A material's microstructure can include voids orinclusions which will lead to a non-uniform stress and strain state inthe microstructure. Multiscale modeling couples the microstructuralstress state with the macroscopic calculation to capture the effects ofthe microstructure in a macroscopic calculation.

The virtual internal work density at a macroscopic material point can bewritten as

$\begin{matrix}{{{\delta \; W^{int}} = {{\sigma_{M}\text{:}{\delta ɛ}_{M}} = {\frac{1}{\Omega }{\int_{\Omega}{\sigma_{m}\text{:}{\delta ɛ}_{m}d\; \Omega}}}}},} & \left( {4\text{-}1} \right)\end{matrix}$

where |Ω| is the volume of the microstructural RVE, and σ_(m) and δε_(m)are the microstructural stress and virtual strain, respectively. Themacroscopic stress σ_(m) and macroscopic virtual strain δε_(m) can becomputed using the Hill-Mandel Lemma to integrate these quantities overthe microstructural RVE as

$\begin{matrix}{\sigma_{M} = {{\frac{1}{\Omega }{\int_{\Omega}{\sigma_{m}d\; \Omega \mspace{14mu} {and}\mspace{14mu} {\delta ɛ}_{m}}}} = {\frac{1}{\Omega }{\int_{\Omega}{{\delta ɛ}_{m}d\; {\Omega.}}}}}} & \left( {4\text{-}2} \right)\end{matrix}$

In this exemplary example, the subscript m represents the microscopicquantities, and the subscript M represents the macroscopic homogenizedquantities.

For an elasto-plastic material, the constitutive law in themicrostructure can be written as

σ_(m) =C _(m):ϵ_(m) ^(el) =C _(m):(ε_(m)−ε_(m) ^(pl))  (4-3)

where C_(m) is the microscopic elastic stiffness tensor, and ε_(m) ^(el)and ε_(m) ^(pl) are the microscopic elastic strain and plastic strain,respectively. The total strain ε_(m) at a point within themicrostructure is given by

ε_(m) =∫dε _(m)  (4-4)

where dε_(m) is the microstructural strain increment due to anassociated macroscopic strain increment δε_(m), applied as boundaryconditions on the microstructural RVE. According to the Hill-Mandellemma, the homogenized stress σ_(m) can be computed by averaging themicrostructural stress σ_(m) in the RVE. Through this homogenizationprocess, the mechanical response at each macro material/integrationpoint is coupled with a microscale RVE model. Meanwhile, the computedmicroscopic history-dependent internal variables are stored in the RVEfor continued analysis. This concurrent framework is advantageous sincethe constitutive law can be adjusted on the fly based on themicrostructural characteristics. The multiscale framework obviates theneed for a cumbersome equation-based phenomenological constitutive lawto account for the behavior of history-dependent material with complexmicro-morphologies and nonlinear behavior, such as plasticity.

Strain Softening and Damage

The homogenization scheme may encounter stability problems when thematerial experiences strain-softening, such as material failuredescribed by a progressive damage model. For an elasto-plastic materialwith damage, the damaged microstructural stress can be written as

σ_(m) ^(d)=(1−d _(m))C _(m): ε_(m) ^(el)=(1−d _(m))C _(m):(ε_(m)−ε_(m)^(pl)),  (4-5)

where d_(m) is a non-decreasing scalar damage parameter describing theirreversible isotropic damage process. Anisotropic damage can also beconsidered, but the scalar damage parameter needs to be replaced by atensor. This damage parameter acts on the stress of a referenceelasto-plastic material,

σ_(m) ⁰ =C _(m): ε_(m) ^(el) =C _(m):(ε_(m)−ε_(m) ^(pl)),  (4-6)

which gives

σ_(m) ^(d)=(1−d _(m))σ_(m) ⁰  (4-7)

This reference elasto-plastic material stores the history-dependentstate variables and provides the relationship between the elastic strainε_(m) ^(pl) and plastic strain ε_(m) ^(pl). The damage parameter can bewritten as a function of state variables q_(m) (e.g., the effectiveplastic strain),

d _(m) =d _(m)(q _(m)).  (4-8)

The progressive damage model directly applied to the microstructural RVEsuffers from material instability due to non-physical strain softening,leading to results in which the localization occurs in very narrow bandsof elements. Since no regularization is introduced in this example, thewidth of the band depends on the mesh size. Even if the microscale RVEis regularized with a physical band width, the homogenized macroscopicstress-strain response still strongly depends on the RVE size, so thatit cannot represent the material behavior at local macroscopic materialpoint.

Stabilized Micro-Damage Algorithm

A micro-damage stabilization algorithm is proposed which removes thematerial strain softening instability associated with traditional damagemodels. This is accomplished by introducing a reference elasto-plasticRVE and decoupling the damage from the plastic evolution. The effectivemacroscopic material law with damage is written as

σ_(M) ^(d) =C _(M) ^(d): ε_(M) ^(el) =C _(M) ^(d):(ε_(M)−ε_(M)^(pl)),  (4-9)

where σ_(M) ^(d) is the macroscopic stress of the damaged RVE, and C_(M)^(d) is the macroscopic effective elastic stiffness tensor of thedamaged RVE. Moreover, ε_(M) ^(el) and ε_(M) ^(pl) are the macroscopiceffective elastic strain and plastic strain, respectively. The effectivematerial law of the reference elasto-plastic RVE can be written as

σ_(M) ⁰ =C _(M):ε_(M) ^(el) =C _(M):(ε_(M)−ε_(M) ^(pl)),  (4-10)

where σ_(M) ⁰ is the macroscopic stress of the undamaged reference RVE,whose effective elastic stiffness tensor is denoted by C_(M). Assumingthe damaged elasto-plastic RVE has the same macroscopic effectiveelastic strain (or effective plastic strain), the following relationshipbetween σ_(M) ^(d) and σ_(M) ⁰ exists

σ_(M) ^(d) =C _(M) ^(d):[(C _(M))⁻¹:σ_(M) ⁰].  (4-11)

In a general multi-dimensional heterogeneous material, the macroscopiceffective plastic strain ε_(M) ^(pl) is not necessarily a volume averageof the microscopic plastic strain ε_(m) ^(pl). According to thedefinition of the macroscopic material law in Eq. (4-10), σ_(M) ⁰ shouldvanish if ε_(M) coincides with ε_(M) ^(pl)

σ_(M)=0, if ε_(M)=ε_(M) ^(pl) (or ε_(M) ^(el)=0).  (4-12)

The stabilized micro-damage algorithm computes the effective macroscopicquantities in Eqs. (4-9) and (4-10), as illustrated in FIG. 38.

A strain increment at a macroscopic material point is passed to thereference elasto-plastic RVE as a homogenized strain increment, dε_(M),and applied as a boundary condition. The microscopic material law in theRVE is

σ_(m1) =C _(m):(ε_(m1)−ε_(m) ^(pl)).  (4-13)

The microstructural stress and total strain are computed, and themicrostructural stress is homogenized using the Hill-Mandel lemma toobtain the macroscopic stress

$\begin{matrix}{\sigma_{M}^{0} = {\frac{1}{\Omega }{\int_{\Omega}{\sigma_{m\; 1}d\; {\Omega.}}}}} & \left( {4\text{-}14} \right)\end{matrix}$

The macroscopic effective elastic strain ε_(M) ^(el) (or plastic strainε_(M) ^(pl)) is then separated from the total strain. According to Eq.(4-10), the effective macroscopic elastic strain can be expressed as

ε_(M) ^(el)=(C _(M))⁻¹:σ_(M) ⁰.  (4-15)

This is equivalent to applying the homogenized stress from the referenceRVE, σ_(M) ⁰, to a second RVE which is identical to the first RVE,except that the material is an undamaged elastic material,

σ_(m2) =C _(m):ε_(m2).  (4-16)

The microstructural strain in the second RVE is computed and homogenizedto yield the macroscopic elastic strain as

$\begin{matrix}{ɛ_{M}^{el} = {\frac{1}{\Omega }{\int_{\Omega}{ɛ_{m\; 2}d\; {\Omega.}}}}} & \left( {4\text{-}17} \right)\end{matrix}$

Finally, the macroscopic stress of the damaged material can be computedbased on Eq. (4-9),

σ_(M) ^(d) =C _(M) ^(d):ε_(M) ^(el).  (4-18)

This is equivalent to applying the homogenized strain from the secondRVE, ε_(M) ^(el), to a third RVE with damaged elastic materialproperties,

σ_(m3)=(1−d _(m))C _(m):ε_(m3),  (4-19)

where the microscopic damage parameter, d_(m), is a function of thestate variables, q_(m1), in the first RVE or the referenceelasto-plastic RVE,

d _(m) =d _(m)(q _(m1)).  (4-20)

The microscopic stress in the third RVE is computed and homogenized as

$\begin{matrix}{\sigma_{M}^{d} = {\frac{1}{\Omega }{\int_{\Omega}{\sigma_{m\; 3}d\; {\Omega.}}}}} & \left( {4\text{-}21} \right)\end{matrix}$

The homogenized stress, σ_(M) ^(d), is returned to the macroscale modelas the damaged macroscopic stress at the material point corresponding tothe macroscopic strain.

Effective Damage Parameter

To characterize the damage state of the RVE, an effective macroscopicdamage parameter, d_(M), can be defined as

$\begin{matrix}{d_{M} = {1 - \frac{{\sigma_{M}^{d}\text{:}\sigma_{M}^{0}}}{{\sigma_{M}^{0}\text{:}\sigma_{M}^{0}}}}} & \left( {4\text{-}22} \right)\end{matrix}$

The macroscale homogenized material and the associated RVE are said tobe fully damaged when d_(M)=1. Note that the effective damage parameteris not defined as the field average of the local damage parameter, sofull damage can be achieved even if only part of the RVE is fullydamaged. The effective damage parameter is a natural by-product of thehomogenization scheme and does not affect the stress-strain relation,but it does provide a useful state variable/indicator for trackinglocalization in the macroscale simulation.

Non-Local Macroscopic Damage

A non-local macroscale damage model is adopted to deal with thepathological mesh dependence due to strain softening. The macroscopicdamage is based on the microscale damage computed using the proposedmicro-damage algorithm from the above section. The algorithm capturesthe complex damage mechanisms due to material heterogeneities at themicrostructural level without predefining the form or requiring a lengthscale. This is achieved at a local macroscopic point by homogenizing thedamaged stress computed in the microscale RVE. Although the microscaledamage model does not require a non-local length scale, the macroscalemodel is still subject to the pathological mesh dependence when thehomogenized results from the RVE are passed back to the macroscalematerial point.

A non-local macroscopic length scale is introduced via a convolutionintegral. The non-local microscale damage parameter, d _(m), at point xinside the RVE is obtained as a weighted average over a spatialneighborhood of the macroscale point ξ under consideration,

{tilde over (d)} _(m)(x,ξ)=∫_(B)ω(∥ξ−ξ′∥)d _(m)(x,ξ′)dξ′,  (4-23)

where d_(m)(x, ξ′) is the local damage increment at a microscale point xinside the RVE associated with macroscale point ξ′. Note that thenon-local regularization can also be performed on other variables, suchas strain, stress and effective plastic strain.

The clustering domain decomposition utilized for the reduced order SCAmethod leads to a discrete form of the convolution integral. Thenon-local damage parameter in the I-th cluster {tilde over (d)}_(m) ^(I)at point ξ of a reduced order RVE can be written as

{tilde over (d)} _(m) ^(I)(ξ)=∫_(B)ω(∥ξ−ξ′∥)d _(m) ^(I)(ξ′)dξ^(′),  (4-24)

where d_(m) ^(I)(ξ′) is the local damage parameter in the I-th clusterof the reduced RVE at point ξ′.

The non-local weighting function ω(∥ξ−f ξ′∥) is normalized to preserve auniform field,

$\begin{matrix}{{{\omega \left( {{\xi - \xi^{\prime}}} \right)} = \frac{\omega_{\infty}\left( {{\xi - \xi^{\prime}}} \right)}{\int_{B}{{\omega_{\infty}\left( {{\xi - \xi^{\prime}}} \right)}d\; \xi^{\prime}}}},} & \left( {4\text{-}25} \right)\end{matrix}$

where B denotes the macroscale support domain for the non-localintegration. One possible weighting function, which is utilized for theexamples in this paper, is a polynomial bell function with compactsupport,

$\begin{matrix}{{{\omega_{\infty}(r)} = {\langle{1 - \frac{4r^{2}}{l_{0}^{2}}}\rangle}^{2}},} & \left( {4\text{-}26} \right)\end{matrix}$

where the Macauley brackets

. . .

are defined as

x

=max(x,0), and l₀ is the macroscale length scale parameter. Sinceω_(∞)(r) vanishes for r>l₀/2, the support domain B is circular intwo-dimensions, or spherical in three-dimensions, with a radius l₀/2.

The macroscale length parameter, l₀, determines the width of the damagelocalization band, and limits localization to ensure numericalconvergence to a physically meaningful solution. The non-localmicroscale damage parameter {tilde over (d)}_(m) (or {tilde over(d)}_(m) ^(I)) is utilized in the third RVE, resulting in a microscalestress-strain relation in Eq. (4-19) of

σ_(m3)=(1−{tilde over (d)} _(m))C _(M):ε_(m3).  (4-27)

For the new micro-damage homogenization method with the non-localformulation, the non-local microscale damage parameter {tilde over(d)}_(m) is computed in Step 2 in the algorithm of the self-consistentscheme shown below, and {tilde over (d)}_(m) will affect on themacroscale material responses through homogenization of the RVE stress.

The macroscale length parameter l₀ can be determined by measuring thewidth of the strain localization band, either in high-fidelity directnumerical simulations (DNS) which model the damage evolution processexplicitly, or by experimental image post-processing using digital imagecorrelation (DIC) analysis. For a given weighting function defined byl₀, the mesh size l_(e) of the macroscale model (e.g., FEM) needs to besmall enough to effectively remove the mesh sensitivity and reduce theerror caused by sample aliasing. In this exemplary example, l_(e)<l₀/4is used, and the influence of mesh size will also be investigated in theexample.

Self-Consistent Clustering Analysis: Model Reduction

The concurrent multiscale modeling framework for the micro-damagealgorithm is applicable to complex materials, but additionalcomputational resources are needed to solve the microscale model at eachintegration point. As a result, the whole concurrent simulation isprohibitively expensive for complex microscale representation.

The SCA with a new projection-based self-consistent scheme is proposedto increase the efficiency of the concurrent calculations. Theefficiency of the SCA is achieved via data compression algorithms whichgroup local microstructures into clusters during an offline trainingstage. Grouping material points with similar mechanical behavior intoclusters results in significantly fewer degrees of freedom than theoriginal DNS; the computational speed is thus greatly improved. Theself-consistent scheme introduced in the online stage of the SCAguarantees the accuracy of the reduced order model. Importantly, the SCAis valid for any local nonlinear constitutive law (such as plasticityand damage) of each material phase in the microscale model.

Offline Stage: Mechanistic Material Characterization

Grouping material points with similar mechanical behavior into a singlecluster is performed by domain decomposition of the material pointsusing the clustering methods. First, the similarity between two materialpoints is measured by the strain concentration tensor A_(m)(x), which isdefined as

ε_(m)(x)=A _(m)(x): ε_(m) in Ω,  (4-28)

where ε_(M) is the elastic macroscopic strain corresponding to theboundary conditions on the RVE, and ε_(m)(x) is the elastic local strainat point x in the microscale RVE domain Ω. In a two-dimensional (2D)model, A_(m)(x) has 9 independent components, which can be determined bya set of elastic DNS calculations under 3 orthogonal loading conditions.For a linear elastic material, the strain concentration tensor isindependent of the loading conditions. Other metrics, such as effectiveplastic strain and damage parameter, can also be selected for theoffline data clustering, but may require extra computation in theoffline stage.

After computing the strain concentration tensor A_(m)(x), the k-meansclustering method is used to group data points. Additional details aboutthe clustering algorithm are provided in the end of this example. Sinceall the material points in a cluster are assumed to have the samemechanical responses, the number of the degrees of freedom issignificantly reduced through this compression/clustering step. Thek-means clustering results of a 2D heterogeneous RVE in the matrix phaseand inclusion phase at two levels of resolution are provided in FIG. 39,showing K-means clustering results of the cross-section of atwo-dimensional RVE with circular inclusions. The high-fidelity RVE isdiscretized by a 1200×1200 mesh. Each cluster contains all the separatesub-domains with the same color.

A primary assumption associated with the domain decomposition is thatany local variable β(x) is uniform within each cluster. Globally, thisis equivalent to having a piece-wise uniform profile of the variable inthe RVE. This piecewise uniform approximation enables us to reduce thenumber of degrees of freedom for the Lippmann-Schwinger equation, whichis solved in the following online stage. Meanwhile, the so-calledinteraction tensor D^(IJ) can also be extracted as an invariant insidethe reduced Lippmann-Schwinger equation.

It should be noted that although the domain decomposition is based on aspecific selection of elastic properties for each material phase in theoffline stage, the same database can be used for predicting responsesfor new combinations of material constituents in the online stage. Eventhough the absolute value of the strain concentration tensor A_(m)(x)strongly depends on the phase properties, the clustering results aremore sensitive to the distribution of A_(m)(x) in the RVE, whichcharacterizes the geometrical heterogeneity. In this exemplary example,the same database shown in FIG. 4-4 is used for predicting the responsesof two different RVEs embedded with hard or soft inclusions.

Online Stage: A New Projection-Based Self-Consistent Scheme

The equilibrium condition in the RVE can be rewritten as a continuousLippmann-Schwinger integral equation by introducing a homogeneousisotropic linear elastic reference material,

Δε_(m)(x)+∫_(Ω)Φ⁰(x,x′):[Δσ_(m)(x′)−C ⁰: Δε_(m)(x′)]dx′−Δε ⁰=0,  (4-29)

where Δε⁰ is the far-field strain increment controlling the evolution ofthe local strain. The far-field strain is uniform in the RVE. Thereference material is isotropic linear elastic, and its stiffness tensorC⁰ can be determined by two independent Lame parameters λ⁰ and μ⁰,

C ⁰ =f(λ⁰,μ⁰)=λ⁰ I⊗I+μ ⁰ II,  (4-30)

where I is the second-rank identity tensor, and II is the symmetric partof the fourth-rank identity tensor. Moreover, Δε_(m)(x) and Δσ_(m)(x)are the microscopic strain and stress increments, respectively.Averaging the incremental integral equation in Eq. (29) in Ω leads to

$\begin{matrix}{{{\frac{1}{\Omega }{\int_{\Omega}{{{\Delta ɛ}_{m}(x)}{dx}}}} + {\frac{1}{\Omega }{\int_{\Omega}{\left\lbrack {\int_{\Omega}{{\Phi^{0}\left( {x,x^{\prime}} \right)}{dx}}} \right\rbrack {\text{:}\left\lbrack {{{\Delta\sigma}_{m}\left( x^{\prime} \right)} - {C^{0}\text{:}{{\Delta ɛ}_{m}\left( x^{\prime} \right)}}} \right\rbrack}{dx}^{\prime}}}} - {\Delta ɛ}^{0}} = 0.} & \left( {4\text{-}31} \right)\end{matrix}$

Using the boundary conditions for deriving the Green's function, Eq.(4-30) can be equivalently written in the spatial domain as

∫_(Ω)Φ⁰(x,x′)dx=0.  (4-32)

Substituting Eq. (4-32) into (4-31) gives

$\begin{matrix}{{{\Delta ɛ}^{0} = {\int_{\Omega}{{{\Delta ɛ}_{m}(x)}{dx}}}},} & \left( {4\text{-}33} \right)\end{matrix}$

which indicates that the far-field strain increment is always equal tothe ensemble averaged strain increment in the RVE. To compute the strainincrement Δε_(m)(x) in the integral Eq. (4-29), constraints are neededfrom the macroscopic boundary conditions. For the homogenization schemeproposed above, two types of constraints are used: 1) macro-strainconstraints in RVEs 1 and 3

$\begin{matrix}{{{\frac{1}{\Omega }{\int_{\Omega}{{{\Delta ɛ}_{m}(x)}{dx}}}} = {{{\Delta ɛ}_{m}\mspace{14mu} {or}\mspace{14mu} \Delta \; {ɛ_{m}}^{0}} = {\Delta ɛ}_{m}}};} & \left( {4\text{-}34} \right)\end{matrix}$

and 2) a macro-stress constraint in RVE 2

$\begin{matrix}{{\frac{1}{\Omega }{\int_{\Omega}{{\sigma_{m}(x)}{dx}}}} = {\sigma_{M}.}} & \left( {4\text{-}35} \right)\end{matrix}$

For more general cases, macro-stress and mixed constraints can also beformulated. Here the boundary conditions are applied as constraints onthe volume averages of strain or stress inside the RVE. This methodologydiffers from the standard finite element method in which the boundaryconditions constrain the displacement or force at the RVE boundaries.

As the full-field calculations (e.g., FFT-based method) of thecontinuous Lippmann-Schwinger equation may require excessivecomputational resources, the discretization of the integral equation isperformed based on the domain decomposition in the offline stage. Withthe piecewise uniform assumption in Eq. (4-29), the number of degrees offreedom and the number of the internal variables in the new system canbe reduced. After decomposition, the discretized integral equation ofthe I-th cluster can be derived as

Δε_(m) ^(I)+Σ_(j=1) ^(k) D ^(IJ):[Δσ_(m) ^(J) −C ⁰:Δε_(m)^(J)]−Δε⁰=0,  (4-36)

where Δε_(m) ^(J) and Δσ_(m) ^(J) are the microscopic strain and stressincrement in the J-th cluster. The interaction tensor, D^(IJ), isdefined in Eq. (4-36), and is related to the Green's function of thereference material. After the discretization, the far field strain isstill equal to the average strain in the RVE,

Δε⁰=Σ_(l=1) ^(k) c ^(I)Δε_(m) ^(I),  (4-37)

where c^(I) is the volume fraction of the I-th cluster. Meanwhile themacroscopic boundary conditions are also required to be discretized. Forinstance, the discrete form of the macro-strain constraint can bewritten as

E _(l=1) ^(k) c ^(I)Δε_(m) ^(I)=Δε_(M) or Δε⁰=Δε_(M).  (4-38)

Similarly, the discretized macro-stress constraint becomes

Σ_(I=1) ^(k) c ^(I)Δσ_(m) ^(I)=Δσ_(M).  (4-39)

An important feature of the continuous Lippmann-Schwinger in Eq. (4-29)is that its solution is independent of the choice of the referencematerial C⁰. This can be explained by the fact that the physical problemis fully described by the equilibrium condition and the prescribedmacroscopic boundary conditions. However, once the equation isdiscretized based on the piecewise uniform assumption, the equilibriumcondition is not strictly satisfied at every point in the RVE, and thesolution of the reduced system depends on the choices of C⁰. Thisdiscrepancy can be reduced by increasing the number of clusters into thesystem, but with a computational cost increase due to the increaseddegrees of freedom.

To achieve both efficiency and accuracy, a self-consistent scheme isused in the online stage, which shows good accuracy with fewer clusters.In the self-consistent scheme, the stiffness tensor of the referencematerial C⁰ is set approximately the same as the homogenized stiffnesstensor C,

C ⁰ →C.  (4-40)

The homogenized stiffness tensor C of the RVE can be expressed as

C=Σ _(I=1) ^(k) c ^(I) C _(alg) ^(I) :A _(m) ^(I),  (4-41)

where C_(alg) ^(I) is the algorithm stiffness tensor of the material inthe I-th cluster and is an output of the local constitutive law for thecurrent strain increment in the cluster,

$\begin{matrix}{C_{alg}^{I} = {\frac{{\partial\Delta}\; \sigma_{m}^{I}}{{\partial\Delta}\; ɛ_{m}^{I}}.}} & \left( {4\text{-}42} \right)\end{matrix}$

The strain concentration tensor of the I-th cluster A_(m) ^(I) relatesthe local strain increment in the I-th cluster Δε_(m) ^(I) to thefar-field strain increment Δε⁰,

Δε_(m) ^(I) =A _(m) ^(I):Δε⁰.  (4-43)

The strain concentration tensor A_(m) ^(I) can be determined by firstlinearizing the discretized integral equation (4-36) using C_(alg) ^(I)and then inverting the Jacobian matrix of the Newton's method. Since Cis only required for the self-consistent scheme, the calculation of C isonly performed once after the convergence of the Newton's method to savethe computational cost.

Due to the nonlinearity of the material responses, such as plasticity,it is usually not possible to determine an isotropic C⁰ which providesan exact match with C. The self-consistent scheme is formulated as anoptimization problem, where the optimum isotropic C⁰ minimizes the errorbetween the predicted average stress increments. Although this schemedoes not require computing C explicitly, it has mainly two drawbacks.First, the optimization problem is under-determined under pure shear orhydrostatic loading conditions, so that one of two independent elasticconstants need to be estimated. More importantly, the modulus of theoptimum reference material may become negative under complex loadingconditions, which is deleterious to the convergence of the fixed-pointmethod.

In this exemplary example, a new self-consistent scheme is proposedbased on projection of the effective stiffness tensor C, which isformulated as two well-defined optimization problems. For a 2D planestrain problem, the stiffness tensor of the isotropic reference materialC_(pe) ⁰ is decomposed as

C _(iso) ⁰=2κ⁰ J+2μ⁰ K,  (4-44)

where the 2D bulk modulus can be related to the Lamé parameters,

κ⁰=λ⁰+μ⁰.  (4-45)

The fourth-rank tensor J and K are defined as

J=½(I⊗I) and K=II−J,  (4-46)

where I is the second-rank identity tensor, II is the fourth-ranksymmetric identity tensor. It can be shown that J and K are orthogonalto each other,

J::K=0,  (4-47)

and

J::J=1, K::K=2,  (4-48)

where “::” denotes the inner product between two fourth-order tensors,or A::B=A_(ijkl)B_(ijkl).

To find the optimum κ⁰ and μ⁰ from the projection of the effectivestiffness tensor C, two optimization problems are defined

$\begin{matrix}{{\kappa^{0} = {\underset{\{\kappa^{\prime}\}}{argmin}{{\left\lbrack {C - {C_{iso}\left( \kappa^{\prime} \right)}} \right\rbrack \text{:}\Delta \; ɛ_{h}^{0}}}^{2}}}{and}} & \left( {4\text{-}49} \right) \\{{\mu^{0} = {\underset{\{\mu^{\prime}\}}{argmin}{{\left\lbrack {C - {C_{iso}\left( \mu^{\prime} \right)}} \right\rbrack \text{:}\Delta \; ɛ_{d}^{0}}}^{2}}},} & \left( {4\text{-}50} \right)\end{matrix}$

where Δε_(h) ⁰ and Δε_(d) ⁰ are the hydrostatic and deviatoric parts ofthe far-field strain increment Δε⁰. By taking the derivative of the costfunctions in Eqs. (4-49) and (4-50) for finding the stationary points,the optimum κ⁰ and μ⁰ can be expressed in Voigt notation as

$\begin{matrix}{\mspace{20mu} {{\kappa^{0} = \frac{c_{11} + c_{12} + c_{21} + c_{22}}{4}}\mspace{20mu} {and}}} & \left( {4\text{-}51} \right) \\{{\mu^{0} = {{\eta_{1}\left( \frac{c_{11} - c_{12} - c_{21} + c_{22}}{4} \right)} + {\left( {1 - \eta_{1}} \right)C_{33}} + {\eta_{2}\left( {C_{13} - C_{23} - C_{32} + C_{31}} \right)}}},\mspace{20mu} {with}} & \left( {4\text{-}52} \right) \\{\mspace{20mu} {{\eta_{1} = \frac{\left( {{\Delta \; ɛ_{11}^{0}} - {\Delta ɛ}_{22}^{0}} \right)^{2}}{\left( {{\Delta \; ɛ_{11}^{0}} - {\Delta \; ɛ_{22}^{0}}} \right)^{2} + \left( {\Delta \; \gamma_{12}^{0}} \right)^{2}}},\mspace{20mu} {\eta_{2} = {\frac{\left( {{\Delta \; ɛ_{11}^{0}} - {\Delta ɛ}_{22}^{0}} \right){\Delta\gamma}_{12}^{0}}{\left( {{\Delta ɛ}_{11}^{0} - {\Delta ɛ}_{22}^{0}} \right)^{2} + \left( {\Delta\gamma}_{12}^{0} \right)^{2}}.}}}} & \left( {4\text{-}53} \right)\end{matrix}$

For cases when the denominator (Δε₁₁ ⁰−Δε₂₂ ⁰)²+(Δγ₁₂ ⁰)² vanishes,η₁=0.5 and η₂=0 are used. Specifically, if the effective macroscopicmaterial is orthotropic, the third term in Eq. (4-52) can be dismissed.Similarly, this self-consistent scheme can also be extended to 3Dmaterials. The expressions of κ⁰ and μ⁰ for the 3D self-consistentscheme are derived based on Eq. (4-49) and (4-50).

$\begin{matrix}{{\kappa^{0} = \frac{c_{11} + c_{12} + c_{13} + c_{21} + c_{22} + c_{34} + c_{31} + c_{32} + c_{33}}{9}}{and}} & \left( {4\text{-}54} \right) \\{{\mu^{0} = \frac{\Delta \; ɛ_{d}^{0}\text{:}C\text{:}\Delta \; ɛ_{d}^{0}}{\Delta \; ɛ_{d}^{0}\text{:}T\text{:}\Delta \; ɛ_{d}^{0}}},} & \left( {4\text{-}55} \right)\end{matrix}$

where T is a transformation tensor derived from the optimum problem. Itcan be written in Voigt notation as

$\begin{matrix}{T = {\begin{Bmatrix}{4/3} & {{- 2}/3} & {{- 2}/3} & 0 & 0 & 0 \\{{- 2}/3} & {4/3} & {{- 2}/3} & 0 & 0 & 0 \\{{- 2}/3} & {{- 2}/3} & {4/3} & 0 & 0 & 0 \\0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & 0 & 0 & 1 & 0 \\0 & 0 & 0 & 0 & 0 & 1\end{Bmatrix}.}} & \left( {4\text{-}56} \right)\end{matrix}$

Eqs. (4-52) and (4-55) show that the optimum shear modulus, μ⁰, isweighted by the loading direction, which helps to capture the hardeningeffect on the stiffness tensor. This improves the accuracy of theself-consistent scheme, as compared to simply determining κ⁰ and μ⁰ byminimizing the distance between the stiffness tensors C_(iso) ⁰ and C,which would correspond to averaging the optimum shear modulus in alldirections.

The algorithm of the self-consistent scheme comprises the followingsteps.

-   -   1. Initial conditions and initialization: set (κ⁰, μ⁰); n=0;        {Δε_(m) ^(I), Δε⁰}_(n)=0;    -   2. For loading increment n+1, update the interaction tensor        D^(IJ) and the stiffness tensor of the reference material C⁰        based on (κ⁰, μ⁰);    -   3. Solve the discretized Lippmann-Schwinger equation (4-36). For        nonlinear materials, Newton's method is used;    -   4. Compute the effective stiffness tensor C, and calculate the        optimum (κ⁰, μ⁰) using Eqs. (4-51) and (4-52) (2D plane strain);    -   5. Check error criterion for (κ⁰, μ⁰); if not met, go to step 2;    -   6. Update the strain and stress increments: {Δε_(m)        ^(I),Δε⁰}_(n+1), {Δσ_(m) ¹}_(n+1);    -   7. Update the index of loading increment: n←n+1; and    -   8. If simulation not complete, go to step 2.

For nonlinear materials, the stress increment Δσ_(m) ^(J) is a nonlinearfunction of its strain increment Δε_(m) ^(J), and Newton's method isnormally used to solve the nonlinear system iteratively at each loadincrement.

Microscale Elasto-Plastic RVE with Damage

Numerical examples are presented to investigate and demonstrate theperformance of the micro-damage algorithm combined with an SCA reducedorder model. A detailed study is performed at the microscale RVE levelto investigate the effect of RVE size, and to validate the RVEpredictions from the SCA model against high-fidelity direct numericalsimulations (DNS) with the finite element method and the FFT-basedmethod using a fine mesh and periodic boundary conditions. Thecomputational efficiency of the proposed method is also discussed.

Material Properties and Damage Parameters

The SCA is applied for the homogenization analysis of a nonlinearelasto-plastic heterogeneous material with damage under 2D plane strainconditions. As shown in FIG. 40, an RVE is created with multipleidentical circular inclusions (phase 2) embedded in the matrix (phase1). The volume fraction of the inclusion phase is 30%. In practice, themicrostructural morphology of the RVE can be obtained from imagingtechniques (e.g., scanning electron microscope and focus ion-beam) orcomputational reconstruction. For SCA results comparison and validation,the problem is initially solved using a 1200×1200 DNS model with sidesof length 2L and periodic boundary conditions.

To develop the material clustering database in the offline stage of SCA,linear elastic material properties are used for the phase 1 and phase 2materials:

E _(i)=100 GPa, v _(i)=0.3; E ₂=500 GPa, v ₂=0.19.  (4-57)

where the subscripts 1 and 2 refer to the matrix phase and inclusionphase, respectively. The clustering results based on the materialproperties in Eq. (4-57) are shown in FIG. 39. The number of phase 1clusters is denoted as k₁, and the number of phase 2 clusters is denotedas k₂. The ratio between k₁ and k₂ is defined according to the phasevolume fractions. For the composite material used in this paper, thenumber of clusters in phase 2 is chosen as approximately half the numberof clusters in phase

${1\left( {k_{2} = \left\lbrack \frac{k_{1}}{2} \right\rbrack} \right)},$

since the volume traction of phase 2 is 30%. It will be shown above thatit is possible to use the same material database with the same RVEgeometry but different phase properties.

After forming the material database, different material properties forthe constituent materials can be evaluated. The matrix phase is modeledas an elasto-plastic material which undergoes damage. Its localconstitutive law is given in Eq. (4-5) and an associative plastic flowlaw with a von Mises yield surface is considered. The yield stress cryis determined by the hardening law as a function of the effectiveplastic strain ε ^(pl), which is a monotonically increasing internalstate variable of the plastic material during the deformation. Theyielding stress σ_(Y) ⁰ is equal to 0.5 GPa. The hardening law isconsidered to be piecewise linear and isotropic,

$\begin{matrix}{{\sigma_{Y}\left( {\overset{\_}{ɛ}}^{pl} \right)} = \left\{ {\begin{matrix}{0.5 + {5{\overset{\_}{ɛ}}^{pl}}} & {{\overset{\_}{ɛ}}^{pl} \in \left\lbrack {0,0.04} \right)} \\{0.7 + {2{\overset{\_}{ɛ}}^{pl}}} & {{\overset{\_}{ɛ}}^{pl} \in \left\lbrack {0.04,\infty} \right)}\end{matrix}{{GPa}.}} \right.} & \left( {4\text{-}58} \right)\end{matrix}$

The inclusions remain as a linear elastic material, but either hard orsoft inclusions can be considered by making the Young's modulus of theinclusions either harder or softer than the matrix phase. For these twocases, the properties of the inclusion phase evaluated are

E ₂=500 GPa, v ₂=0.19 hard inclusions,  (4-59)

and

E ₂=1 GPa, v ₂=0.19 soft inclusions.  (4-60)

Damage evolution is modeled as an exponential function of the effectiveplastic strain,

$\begin{matrix}{{d_{m}\left( {\overset{\_}{ɛ}}^{pl} \right)} = \left\{ \begin{matrix}0 & {{{if}\mspace{14mu} {\overset{\_}{ɛ}}^{pl}} \leq {\overset{\_}{ɛ}}^{- {cr}}} \\{1 - {\frac{{\overset{\_}{ɛ}}^{cr}}{{\overset{\_}{ɛ}}^{pl}}{\exp \left( {- {\alpha \left( {{\overset{\_}{ɛ}}^{pl} - {\overset{\_}{ɛ}}^{cr}} \right)}} \right)}}} & {{{if}\mspace{14mu} {\overset{\_}{ɛ}}^{pl}} > {\overset{\_}{ɛ}}^{- {cr}}}\end{matrix} \right.} & \left( {4\text{-}61} \right)\end{matrix}$

where α is an evolution rate parameter and ε ^(cr) is the criticaleffective plastic strain at damage initiation. Since the damage processis irreversible, α≥0. If α=0, the material is purely elasto-plastic withno damage. For positive α, a fully damaged condition (d_(m)=1) will beachieved at a finite effective plastic strain. Unless otherwise stated,the baseline damage properties used in this section are α=100 and ε^(cr)=0.02. The material parameters of the RVE are summarized in Table4-2. nnil Material parameters of the microscale elasto-plastic RVE withdamage. nnil

TABLE 4-1 Material parameters of the microscale elasto-plastic RVE withdamage. E2 E2 E₁ (hard) (soft) (GPa) ν₁ (GPa) (GPa) ν₂ σ_(γ) ⁰(GPa) α(default) ε ^(cr) (default) 100 0.3 500 0.3 0.5 100 0.02

TABLE 4-2 Comparison of the toughness UT obtained using DNS for RVEsizes L/2, L, and 2L. Value in parenthesis indicates relative differenceto the corresponding prediction from RVE with size 2L. Toughness unitsare GJ/m³. RVE Hard inclusions Soft inclusions Size Uniaxial BiaxialShear Uniaxial Biaxial Shear 2L 0.0735 0.3072 0.0103 0.0046 0.00490.0068 L 0.0749_((1.9%)) 0.3184_((3.6%)) 0.0104_((1.0%)) 0.0044_((4.3%))0.004903_((0.5%)) 0.0070_((2.9%)) L/2 0.08²9_((12.7%)) 0.3240_((5.4%))0.0111_((78%)) 0.0056_((21.7%)) 0.0060_((22.4%)) 0.0075_((10.3%))

RVE Size and the Stabilized Micro-Damage Algorithm

The effect of RVE size on the homogenized results is evaluated underuniaxial tension, biaxial tension, and shear loading conditions. Acomparison is made between an RVE of size L/2, an RVE of size L, and anRVE of size 2L (see FIG. 40). Each RVE had the same inclusion radius,volume fraction, and nearest inclusion distance. Three loadingdirections in terms of the macroscopic strain ε_(M) are considered:

1) uniaxial strain condition

{ε_(M)}₁₁=ε, {ε_(M)}₂₂=0, {ε_(M)}₁₂=0;  (4-62)

2) biaxial strain condition

{ε_(M)}₁₁=ε, {ε_(M)}₂₂=ε, {ε_(M)}₁₂=0;  (4-63)

3) shear strain condition

{ε_(M)}₁₁=ε, {ε_(M)}₂₂=ε, {ε_(M)}₁₂=0;  (4-64)

where ε denotes the magnitude of the macroscopic strain.

For each step in the proposed homogenization scheme above, thecomputations are performed using a finite element model with periodicboundary conditions. The homogenized stress-strain curves{σ_(M)}₁₁-{ε_(M)}₁₁ for the damaged RVEs are shown in FIG. 41, where thesubscript 11 indicates the homogenized values in the 11 direction(direction of uniaxial loading). nnil Comparison of the toughness U_(T)obtained using DNS for RVE sizes L/2, L, and 2L. Value in parenthesisindicates relative difference to the corresponding prediction from RVEwith size 2L. Toughness units are GJ/m³.

TABLE 4-3 Computational time of the RVE homogenization of variousnumbers of clusters on one Intel i7-3632 processor. ki 1 4 16 64 256 102DNS time Wall clock 0.23 0.35 0.80 2.5 11.2 117. 7814 Speedup of 3.4 ×10⁴ 2.2 × 10⁴ 9.8 × 10³ 3.1 × 10³ 698 66 NA SCA

TABLE 4-4 Calibrated damage parameters (α, ε ^(cr)) for SCA fordifferent numbers of clusters using energy regularization. The damageparameters are unitless. Hard inclusions Soft inclusions α ε ^(cr) α ε^(cr) k_(l) = 4 52 0.0092 100 0.0015 k_(l) = 8 62 0.0112 60 0.0015 k_(l)= 16 63 0.0120 60 0.0020 k_(l) = 32 67 0.0128 50 0.0020 k_(l) = 64 720.0134 40 0.0020 k_(l) = 128 77 0.0140 30 0.0020 k_(l) = 256 81 0.014630 0.0025 DNS 100 0.02 100 0.02

Inspection of the curves in FIG. 41 reveals some change in themacroscopic responses given by the micro-damage homogenizationalgorithm, but the response with damage is not overly sensitive to theRVE size.

The material toughness U_(T) is defined as the energy per unit volumeabsorbed before material failure due to load in a fixed loadingdirection. In a one-dimensional case, the material toughness is simplythe area under a stress-strain curve. In general, this is expressedmathematically as

U _(T)=∫₀ ^(ε) ^(M) ^(f) σ_(M) ^(D) dε _(M),  (4-65)

where ε_(M) ^(f) is the failure strain for a fully damaged material(d_(M)=0.99 is used as the failure strain in this calculation). Thetabulated results in Table 4-4 show that the relative difference of thematerial toughness decreases with increasing RVE size. The homogenizedresults are not sensitive to the RVE size, which validates the existenceof the RVE. Therefore, it provides an effective material damage modelfor a macroscale material point within a concurrent simulation.

SCA for an RVE with Hard/Soft Inclusions without Material Damage

The SCA reduced order method is first applied to an elasto-plastic RVEwithout considering material damage. The material clustering database isdetermined in the offline stage using hard inclusions.

Three loading directions in terms of the macroscopic strain areconsidered: 1) uniaxial tension, 2) biaxial tension, and 3) shear. Thehomogenized stress strain curves in the 11 direction({σ_(M)}₁₁-{ε_(M)}₁₁) of the undamaged RVE with hard inclusions arepresented in the lefthand side of FIG. 42. The numbers of clustersconsidered in material phase 1 are 1 and 256, and the corresponding DNSresults are plotted as the solid lines for comparison. These resultsshow that the SCA predictions under all the three macro-strainconstraints are quite accurate, even with only a single cluster in phase1.

For soft inclusions, it is not possible to achieve the same accuracywith the same number clusters. The right-hand side of FIG. 42 shows thatk₁=1 is not sufficient to capture the stress-strain relation in theelastic or plastic regimes, especially for the uniaxial and biaxialloading conditions. This observation is consistent with the fact thatthe self-consistent theory in analytical micromechanics methods tends tounderestimate the stiffness of porous materials. Nevertheless, byincreasing the resolution in the RVE, good agreement with the DNSresults can still be achieved, and with significantly fewer degrees offreedom than required by DNS.

Again, it should be emphasized that although the SCA database is createdbased on the RVEs with hard inclusions (see Eq. (4-57)), it is valid forother RVEs with the same morphology but different combinations ofmaterial properties. As a result, the SCA database (clusters and theinteraction tensor D^(IJ)) can be regarded as a microstructural databasewhich characterizes the geometric heterogeneity of the material andenables the online reduced-order calculation.

The computational cost comparison in Table 4-3 shows that a typicaltwo-dimensional DNS calculation with a 1200×1200 mesh and 50 loadingincrements requires 7814 s (≈130 min) on one Intel i7-3632 processor,while the online stage of SCA (programmed in MATLAB) took 0.35 s, 2.5 s,117.8 s on one Intel i7-3632 processor for k₁=4, k₁=64 and k₁=1024,respectively.

TABLE 4-5 Computational time of the 2D concurrent simulations of variousnumbers of clusters in phase 1 on 24 cores. k1 Estimated 4 8 16 32 64DNS time Wall clock time 19 44 117 416 1948 4.2 × 10⁵ (295 (min) days)Speedup of SCA 2.2 × 10⁴ 9.5 × 10³ 3.6 × 10³ 1.0 × 10³ 216 NA

TABLE 4-6 Computational time of the 3D concurrent simulations of variousnumbers of clusters in phase 1 on 72 cores. SCA (k_(l) = 8) EstimatedDNS time Wall clock time (min) 56 h 1.1 × 10⁵ days Speedup of SCA 4.6 ×10⁴ NA

SCA for an RVE with Hard/Soft Inclusions and Material Damage

The SCA reduced order method is next applied to the same elasto-plasticRVE with hard and soft inclusions. The matrix phase of the RVE isallowed to undergo material damage, which is computed using themicro-damage algorithm and the damage evolution law in Eq. (4-61).

The homogenized stress-strain results computed using both DNS and SCAwith material damage are shown in FIG. 43 for hard inclusions and FIG.44 for soft inclusions. Results are presented for uniaxial tensionloading, biaxial tension loading, and shear loading. With the same levelof discretization, the SCA predictions with material damage considereddo not exhibit the same level of accuracy as the SCA predictions withoutconsidering material damage (see FIG. 42). The clustering algorithm wasnoted to be stiffer than the DNS calculation for the number of clustersevaluated. This difference is expected since material damage is a morelocalized process than plasticity, and more clusters are required foradequate resolution in the damaged region. On the other hand, theclustering result based on linear elastic responses may not besufficient for capturing highly localized damage behavior, and thisissue will be investigated in future work.

For hard inclusions, it can be seen that under biaxial tension loadingand shear loading, SCA results matched the peak stresses and damageevolution with very few clusters. For the uniaxial tension loading, theSCA method tends to overpredict the peak stress and damage evolution,meaning that additional clusters may be required. For soft inclusions,the SCA method is capable of capturing the effect of the damage, buttends to overpredict peak stress and damage evolution. This finding isconsistent with the findings in the self-consistent micromechanicstheory. An energy regularization methodology is presented which providesan effective way of calibrating the damage parameters.

The macroscopic material damage behavior can be characterized by thematerial toughness, U_(T), as a function of the strain ratio ε_(M)^(R)/ε_(M) ^(avg), where ε_(M) ^(avg) and ε_(M) ^(R) denote themacroscopic normal strain and shear strain, respectively Due to theisotropy of the RVE considered in this work, the material toughnessshould be uniquely determined by the loading direction denoted by ε_(M)^(R)/ε_(M) ^(avg). For a given set of ε_(M) ^(avg) and ε_(M) ^(R), theloading condition is defined to be

{ε_(M)}₁₁=ε_(M) ^(avg), {ε_(M)}₂₂=ε_(M) ^(avg), {ε_(M)}₁₂=ε_(M)^(R)  (4-66)

The material toughness, U_(T), is defined in Eq. (65). When computingthe material toughness for each loading path, the loading direction, orthe strain ratio ε_(M) ^(R)/ε_(M) ^(avg), is fixed.

FIG. 45 shows the material toughness U_(T) of the RVE as a function ofthe ratio of ε_(M) ^(R) to ε_(M) ^(avg). For hard inclusions, thetoughness decreases monotonically with increasing strain ratio ε_(M)^(R)/ε_(M) ^(avg), and overall effective toughness is higher under thebiaxial loading than by pure shearing loading by more than one order ofmagnitude. Similar to the matrix material, the effective damageevolution of the RVE with hard inclusions is dominated by the deviatoriccomponent of the average stress, resulting in a low toughness under thepure shearing loading. The result for soft inclusions shows that thematerial toughness is no longer a monotonic function of the strainratio, and that the material toughnesses does not vary much with thestrain ratio. This can be explained by the fact that the soft inclusionsundergo volumetric change under hydrostatic loading, which also induceshigh shear deformation and damage in the surrounding matrix material.

Energy Regularization for Calibrating Damage Parameters

The damage model should correctly reflect the energy dissipation in themacroscale fracture process. As a result, parameters in the damageevolution law need to be adjusted according to the measured fractureenergy G_(c) under a given loading direction. Mathematically, thefracture energy should be

G _(c) =G _(c) ^(DNS)(α₀,ε ₀ ^(cr))=G _(c) ^(SCA,k)(α,ε ₀^(cr)),  (4-67)

where k denotes the number of clusters in the SCA reduced-order model.The fracture energy G_(c) represents the dissipated energy due tofracture per unit crack surface in the macroscale, which serves as aphysical material constant and should not depend on the choice ofnumerical models. The units of G_(c) is GJ/m². Given a length parameterl_(h) from the macroscale model, G_(c) can be expressed as

G _(c) =l _(h) U _(F)(α,ε ₀ ^(cr)),  (4-68)

where U_(F) represents the energy dissipated by fracture per unit volumeat a material point. In the paper, the material toughness, U_(T), isused to approximate the fracture energy, U_(F),

G _(c) ≈l _(h) U _(T)(α,ε ₀ ^(cr)),  (4-69)

The fracture energy, G_(c), is usually measured experimentally, but fordemonstration purposes, the DNS results are used as the referencesolution for calibrating the damage evolution rate, α, and the criticalplastic strain, ε ₀ ^(cr), in the SCA model for energy consistency.Since the length parameter, l_(h), is determined by the macroscalemodel, one only needs to match the material toughness between the SCAand DNS RVE models:

U _(T) ^(SCA,k)(α,ε ₀ ^(cr))=U _(T) ^(DNS)(α₀,ε ₀ ^(cr)).  (4-70)

The damage parameters α₀ and ε ₀ ^(cr) used for the DNS calculation areα₀=100 and ε ₀ ^(cr)=0.02. Additionally, it is important to match themaximum stress, σ_(c), during the loading process, which corresponds tothe ultimate strength of the material.

σ_(c) ^(SCA,k)(α,ε ₀ ^(cr))=σ_(c) ^(DNS)(α₀,ε ₀ ^(cr)).  (4-71)

The damage parameters calibrated under uniaxial tension loading areprovided in Table 4-8. The homogenized stress-strain curves followingenergy regularization are shown in FIG. 46. For the material with hardinclusions, it is possible to match the peak stress with very fewclusters, but matching the damage evolution curve requires moreclusters. For the material with soft inclusions, more clusters arerequired to match both the peak stress and the damage evolution curve. Acomparison is provided in FIG. 47 between the damage fields from DNS andSCA following energy regularization. It can be seen that for both hardinclusions and soft inclusions, additional clusters improve thecomputation of the localized damage fields. FIG. 48 shows the materialtoughness calculations following energy regularization for hardinclusions (left) and soft inclusions (right). The results withoutenergy regularization are plotted for reference, and are denoted as(NR).

TABLE 4-7 Summary of the computational time for the RVE homogenizationand concurrent simulations. k denotes the total number of clusters inthe SCA database. 2D RVE 213 concurrent 3D SCA SCA SCA (k = 1536) DNS (k= 96) DNS (est.) (k = 10) DNS (est.) Time 118 s 7814 s 1948 min 4.2 ×10⁵ min 56 h 2.4 × 10⁵ h Speedup 66 NA 216 NA 4.6 × 10⁴ NA

Concurrent Multiscale Results and Discussion

Concurrent multiscale simulations involving strain softening andlocalization are performed under two-dimensional plane strain conditionsand in three-dimensions. The proposed micro-damage homogenization schemecoupled with the SCA method is used. In these examples, the macroscaleproperties are determined by the microstructural morphologies and themicroscale constituents. The SCA material database is compiled duringthe offline stage, greatly reducing the computational cost of analyzingthe microscale RVE model with minimal loss of accuracy, and making theconcurrent simulation computationally feasible. The homogenized materialcan predict the macroscale performance while capturing the physicalphenomena appearing in the microscale.

2D Double-Notched Plate

A 2D tensile specimen with rounded notches in two corners is depicted inFIG. 49. The notches in the opposite corners cause a geometric stressconcentration which will induce localization as the loaded upper edge ofthe specimen is displaced. The simulations are conducted using theexplicit finite element method and 2D plane strain elements with reducedintegration (one integration point per element), and the micro-damagealgorithm with the SCA model is implemented as a user-defined materialat each integration point. Specifically, LU decomposition in the IntelMath Kernel Library (MKL) is utilized for solving the linear systems inthe SCA model. A macroscopic element is deleted when the effectivedamage parameter d_(M) of its associated RVE reaches 0.99.

The effectiveness of the non-local formulation is first investigatedwithout considering energy regularization. The microscale is modeledusing an RVE with hard inclusions; the SCA database with 32 clusters inthe matrix phase is used for the microstructural RVE calculation. FIG.50 presents the load-displacement curves from the concurrent simulationsfor three different macroscale FE meshes shown in FIG. 51. A non-locallength parameter of l₀=2 mm is used. Moreover, the crack patterns afterthe coupon failure for all three mesh sizes are also shown in FIG. 51.It can be concluded that given l₀>4l_(e), the non-local formulation caneffectively diminish the mesh-size dependency, as well as thepathological mesh dependency.

FIG. 52 shows the load-displacement curves for RVEs with hard inclusionsand soft inclusions predicted using concurrent simulation without energyregularization. The numbers of clusters in phase 1 in the microscalereduced-order model ranges from 4 to 64. Since the concurrent simulationwith microscale RVE modeled by DNS is computationally expensive, themultiscale DNS results are not provided for comparison. Fasterconvergence is observed in models with hard inclusions than those withsoft inclusions. In addition, the predictions have a higher fidelity inthe elasto-plastic regime than in the strain-softening regime.Increasing the number of clusters in the microscale RVE clearly improvesthe prediction, but the convergence rate is not satisfactory.

Energy regularization is introduced to increase the fidelity of thecalculation. The calibrated damage parameters for different numbers ofclusters have been provided in Table 4-4. The load-displacement curvesfrom the concurrent simulations after energy regularization are shown inFIG. 53. It can be seen that the accuracy and convergence rate withrespect to the number of clusters have been improved, for both hard andsoft inclusions. Meanwhile, FIG. 54 presents the crack patterns from theconcurrent simulation with energy regularization, which shows aconsistent cracking pattern as the number of cluster is increased.Although the same microstructure and SCA database have been used, thecrack patterns for hard and soft inclusions are noticeably different.The influence of micro-constituent on the macro-responses is captured bythe concurrent simulation driven by the SCA database.

The computational times for the 2D concurrent simulations and k₁ rangingfrom 4 to 64 are listed in Table 4-5. The macroscale mesh size isl_(e)=0.28 mm, and there are in total 8689 elements in the model. Allthe simulations are conducted on 24 cores (in a state-of-the art highperformance computing cluster with the following compute nodes: IntelHaswell E5-2680v3 2.5 GHz 12-cores). A linear scalability with thenumber of cores is observed for small number of clusters. When k₁>16,the computational time of the concurrent simulation increasesapproximately as the square of k₁. Based on the computational time ofSCA (k₁=4) and DNS for the RVE calculation, the estimated DNS time (FE²)of the concurrent simulation would be 295 days (see Table 4-5).

3D Double-Notched Plate

The micro-damage algorithm and the SCA method are further applied to 3Dconcurrent simulations for the double-notched model shown in FIG. 55.The in-plane shape of the 3D model is identical to the 2D example shownin FIG. 49, and the applied boundary conditions are also identical tothe 2D model. The thickness of the FE model is 8 mm. Symmetric boundaryconditions are applied on the back face on the X-Y plane (nodisplacement in the Z-direction), resulting in an effective totalthickness of 16 mm. The non-local length parameter is l₀=2 mm. The meshis refined in the region containing the crack with the characteristicmesh size l_(e)=0.18 mm, so there are approximately 10 elements throughthe thickness of the damage region. For the macroscale FE model, thereare in total 154,371 3D hex elements with reduced integration (oneGauss/integration point per element).

The microscale 3D RVE model has identical spherical inclusions embeddedin the matrix, and the volume fraction of the inclusion phase is 20%.The corresponding DNS FE mesh for the RVE is 80×80×80. In the microscaleSCA model, there are 8 clusters in the matrix phase and 2 clusters inthe inclusion phase. An RVE with hard inclusions is considered in themicroscale. All the material parameters are the same as before (seeTable 4-1). Specifically, the initial damage parameters α=100 and ε ₀^(cr)=0.02 are used. The macroscopic elements will be deleted when theeffective damage parameter d_(M) of the RVE reaches 0.99.

The load-displacement curve from the concurrent simulations with hardinclusions is shown in FIG. 56, and the crack patterns at differentloading states are also provided in FIG. 57. In addition, the microscalestress and damage fields are shown for RVEs at different integrationpoints in the macroscale model. In the concurrent coupling, themicroscale RVE reduced order model not only provides the overallstress-strain responses for the macroscale FE model, but also storeslocal quantities such as plastic strain and damage parameter inside themicrostructure/clusters. It can be observed that the concurrentsimulations are able to demonstrate a non-planar crack through thecross-section due to the constraint effects of the thick coupon. Inaddition, the crack front is curved, as one can see from the snapshotsat displacement Δ=0.15 mm. The damaged region initiates at the center ofthe model cross-section and propagates forward and toward theboundaries.

Due to the limitation of computational resources, convergence tests andenergy regularization have not yet been performed for the 3D problem,but these preliminary concurrent simulations demonstrate the capabilityof the SCA multiscale modeling framework for capturing themicrostructural effect. By using 72 cores (Intel Haswell E5-2680v3 2.5GHz 12-cores), the concurrent simulation with the Intel MKL libraryrequired approximately 56 hours. Based on the time comparison for the 3DRVE computation, the estimated DNS time for the same concurrentsimulation, with the original 80×80×80 FEM mesh in the microscale andthe same computational resources, would be more than 1×10⁵ days (seeTable 4-6).

In this exemplary example, a stable micro-damage algorithm has beenintroduced which removes the need for a material length parameter in themicrostructural RVE. By matching the effective elastic strains of theundamaged and damaged RVE, strain localization is avoided in themicroscale RVE, and the homogenized behavior in the strain-softeningregime is independent of the RVE size. No material length parameter isrequired in the microscale, and the RVE homogenization is able tocapture the driving mechanisms of the damage process.

A macroscopic material length parameter is utilized to alleviate thematerial instability associated with strain softening behavior in themacroscopic calculation. It is demonstrated that the non-local damagemodel with the length parameter can effectively remove the pathologicalmesh dependence.

The SCA provides an efficient computational technique for an RVEundergoing nonlinear history-dependent deformation. A good tradeoffbetween efficiency and accuracy is possible with SCA through the domaindecomposition based on k-means clustering of high-fidelity data and themicromechanics-based self-consistent scheme. A new projection-basedself-consistent scheme for solving the Lippmann-Schwinger equation isproposed in the SCA online/prediction stage, which is able to handle theRVE problem under complex loadings in a concurrent simulation. With thesame SCA database, different macroscale crack paths are predicted forRVEs with different inclusion properties (hard inclusions versus softinclusions). The proposed method is general and can be applied tovarious material systems with more complex settings.

A solution comparison with the DNS has shown that the proposedmicro-damage methodology coupled with the SCA is capable of achievingaccurate solutions at an incredible reduction of computational expense(see Table 4-7). For a 2D plane strain problem, the solution time usingthe SCA is 1948 minutes using 64 clusters compared with an estimated 295days required for a DNS calculation. For a 3D problem, the SCA solutiontime is 56 hours, compared with an estimated 1×10⁵ days using the DNS.

Clustering of the Strain Concentration Tensor

For a 2-dimensional model, the strain concentration tensor A(x) in eachmaterial point has 9 independent components. The format of the raw datafor a 2D material is shown in Table 4-8.

TABLE 4-8 Format of the raw data for a 2D material Data- Index A₁₁ A₂₂A₃₃ A₁₂ A₂₁ A₂₃ A₃₂ A₁₃ A₃₁ 1 — — — — — — — — — 2 — — — — — — — — — . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . N — — — — — — —— —where N is the total number of discretization points in the DNS. For the1200×1200 mesh used in the paper, N is equal to 1440000. For a3-dimensional model, A(x) has 36 independent components.

The next step is to perform the domain decomposition by grouping similardata points using k-means clustering. Mathematically, given a set ofstrain concentration tensors, the objective of k-means clustering is tominimize the within-cluster least squares sum for the k sets S={S¹, S²,. . . , S^(k)} to obtain the shape of the clusters:

$\begin{matrix}{{S = {\underset{S^{\prime}}{argmin}{\sum\limits_{J = 1}^{k}{\sum_{n \in S^{J}}{{A_{n} - A_{J}}}^{2}}}}},} & \left( {4\text{-}72} \right)\end{matrix}$

where A_(n) is the strain concentration tensor of the n-th data point,and A_(j) is the mean of all the strain concentration tensors at thepoints within the cluster S_(j). The above norm is defined as usual,e.g., for a general second-order matrix Z of dimension m×m

∥Z∥=√{square root over (Σ_(i=1) ^(m)Σ_(j=1) ^(m) z _(ij) ²)}=√{squareroot over (trace(Z ^(T) Z))},  (4-73)

and is called the Frobenius norm of matrix Z. In this exemplary example,the standard algorithm is used to solve the k-means clustering problem,which is essentially an optimization process minimizing the sum in Eq.(4-72).

At the initialization step, k data points are randomly selected from thedata set and served as the initial means. The algorithm then iteratesbetween the following two steps,

1. Assignment step: Each data point is assigned to the cluster whosemean is nearest to the data point. In other words, within the t-thiteration, ∀{A_(n)}∈S_(I) ^((t))

∥{A _(n) }−A _(I) ^((t))∥² ≤∥{A _(n) }−A _(J) ^((t))∥² ∀J,J≠1  (4-74)

2. Update step: The mean values of the data points in the new clusterare recalculated for iteration t+1,

$\begin{matrix}{{A_{I}^{(t)} = {\frac{1}{N_{I}^{(t)}}{\sum_{{\{ A_{n}\}} \in S_{I}^{(t)}}\left\{ A_{n} \right\}}}},} & \left( {4\text{-}75} \right)\end{matrix}$

where N_(i) ^((t)) is the number of data points in cluster S_(I) ^((t)).

When the assignment of data points no longer changes, the algorithm issaid to converge to a local optimum. Finally, it should be noted thatother clustering methods can also be applied for this problem, such asthe density-based spatial clustering of applications with noise (DBSCAN)method.

Computing the Interaction Tensor

In the discretized/reduced Lippmann-Schwinger equation, the piecewiseuniform assumption can be utilized to extract the interaction tensorD^(IJ), which represents the influence of the stress in the J-th clusteron the strain in the I-th cluster. In a periodic RVE domain Ω, theinteraction tensor can be written as a convolution of the Green'sfunction and the characteristic functions,

$\begin{matrix}{{D^{IJ} = {\frac{1}{c^{I}{\Omega }}{\int_{\Omega}{\int_{\Omega}{{\chi^{I}(x)}{\chi^{J}\left( x^{\prime} \right)}{\Phi^{0}\left( {x,x^{\prime}} \right)}{dx}^{\prime}{dx}}}}}},} & \left( {4\text{-}76} \right)\end{matrix}$

where c^(I) is the volume fraction of the I-th cluster and |Ω| is thevolume of the RVE domain. Φ⁰ (x, x′) is the fourth-order periodicGreen's function associated with an isotropic linear elastic referencematerial and its stiffness tensor is C⁰. This reference material isintroduced in the online stage as a homogeneous media to formulate theLippmann-Schwinger integral equation. With the periodicity of the RVE,Φ⁰ (x, x′) takes a simple form for isotropic materials in the Fourierspace,

$\begin{matrix}{{{{\hat{\Phi}}^{0}(\xi)} = {{\frac{1}{4\mu^{0}}{{\hat{\Phi}}^{1}(\xi)}} + {\frac{\lambda^{0} + \mu^{0}}{\mu^{0}\left( {\lambda^{0} + {2\mu^{0}}} \right)}{{\hat{\Phi}}^{2}(\xi)}}}},{with}} & \left( {4\text{-}77} \right) \\{{{\hat{\Phi}}_{ijkl}^{1}(\xi)} = {\frac{1}{{\xi }^{2}}\left( {{\delta_{ik}\xi_{j}\xi_{l}} + {\delta_{il}\xi_{j}\xi_{k}} + {\delta_{jl}\xi_{i}\xi_{k}} + {\delta_{jk}\xi_{i}\xi_{l}}} \right)}} & \left( {4\text{-}78} \right) \\{{{{\hat{\Phi}}_{ijkl}^{2}(\xi)} = {- \frac{\xi_{i}\xi_{j}\xi_{k}\xi_{l}}{{\xi }^{4}}}},} & \left( {4\text{-}79} \right)\end{matrix}$

where ξ is the coordinate in Fourier space corresponding to x in realspace, and δ_(ij) is the Kronecker delta function. λ⁰ and μ⁰ are Lameconstants of the reference material. The expression of {circumflex over(Φ)}_(ijkl) ⁰ (ξ) is not well defined at frequency point ξ=0. However,imposing by the boundary conditions for deriving the Green's function, auniformly distributed polarization stress field will not induce anystrain field inside the RVE, which indicates

{circumflex over (Φ)}_(ijkl) ⁰(ξ=0)=0.  (4-80)

Based on Eq. (77), the convolution term in the spatial domain in Eq.(4-76) can be translated into a direct multiplication at each point ξ inthe frequency domain using a Fourier transformation,

Φ _(J) ⁰(x)=∫_(Ω)χ^(J)(x′)Φ⁰(x,x′)dx′=

⁻¹({circumflex over (χ)}^(J)(ξ){circumflex over (Φ)}⁰(ξ)).  (4-81)

It can see from Eq. (4-78) and (4-79) that {circumflex over (Φ)}¹(ξ) and{circumflex over (Φ)}²(ξ) are independent of the material properties,meaning that they can be computed one time only in the offline stage. Ifthe reference material is changed in the self-consistent scheme at theonline stage, only the coefficients relating to its Lame constants inEq. (4-77) need to be updated.

Newton's Iteration in the Online Stage

In the implicit scheme, the residual {r}={r¹; . . . ; r^(k); r^(k+1)} islinearized with respect to {Δε}. Dropping terms of higher order thanlinear gives

$\begin{matrix}{{{\left\{ r \right\} + {\left\{ M \right\} \left\{ {d\; ɛ} \right\}}} = {{0\mspace{14mu} {with}\mspace{14mu} \left\{ M \right\}} = \frac{\partial\left\{ r \right)}{\partial\left\{ {\Delta \; ɛ} \right\}}}},} & \left( {4\text{-}82} \right)\end{matrix}$

where {M} is called the system Jacobian matrix. For I,J=1, 2, . . . , k,

M ^(IJ)=δ_(IJ) I+D ^(IJ):(C _(alg) ^(J) −C ⁰ and M ^(I(k+1))=−I,  (4-83)

where δ_(IJ) is the Kronecker delta in terms of indices I and J, and Iis the fourth-order identity tensor. C_(alg) ^(J) is the so-calledalgorithmic stiffness tensor of the material in the J-th cluster and isan output of the local constitutive law for the current strain incrementin that cluster Δε_(n) ^(J),

$\begin{matrix}{C_{alg}^{J} = {\frac{{\partial\Delta}\; \sigma^{J}}{{\partial\Delta}\; ɛ^{J}}.}} & \left( {4\text{-}84} \right)\end{matrix}$

Although local material damage is decoupled with the equilibriumcondition to avoid material instability in this paper, it is stillpossible to introduce the damage into the local constitutive law andC_(alg) ^(J) can be written as

$\begin{matrix}{{C_{alg}^{J} = {\left( {1 - d^{J}} \right)\frac{{\partial\Delta}\; \sigma_{ud}^{J}}{{\partial\Delta}\; ɛ^{J}}}},} & \left( {4\text{-}85} \right)\end{matrix}$

where ∂Δ_(ud) ^(J)/∂Δε^(J) represents the algorithmic stiffness tensorof the undamaged pure elasto-plastic material. Under the macro-strainconstraint, the remaining components in the system Jacobian matrix are

M ^((k+1)I) =c ^(I) I and M ^((k+1)(k+1))=0.  (4-86)

For macro-stress constraint,

M ^((k+1)I) =c ^(I) C _(alg) ^(I) and M ^((k+1)(k+1))=0.  (4-87)

Finally, the correction of the incremental strain can be expressed as

{dε}=−{M} ⁻¹ {r}  (4-88)

Based on the updated incremental strain, the constitutive relationshipin each cluster can then be used to compute the new incremental stress{Δσ}={Δσ¹; . . . ; Δσ^(k)}. If Δσ^(I) and Δε^(I) have a nonlinearrelation, several iterations are needed so that the residual {r} canvanish.

The foregoing description of the exemplary embodiments of the presentinvention has been presented only for the purposes of illustration anddescription and is not intended to be exhaustive or to limit theinvention to the precise forms disclosed. Many modifications andvariations are possible in light of the above teaching.

The embodiments were chosen and described in order to explain theprinciples of the invention and their practical application so as toactivate others skilled in the art to utilize the invention and variousembodiments and with various modifications as are suited to theparticular use contemplated. Alternative embodiments will becomeapparent to those skilled in the art to which the present inventionpertains without departing from its spirit and scope. Accordingly, thescope of the present invention is defined by the appended claims ratherthan the foregoing description and the exemplary embodiments describedtherein.

What is claimed is:
 1. An integrated process-structure-property modelingframework for design optimization and/or performance prediction of amaterial system, comprising: a database; a plurality of models coupledwith the database, comprising process models and mechanical responsemodels, each model being coupled to the next by passing outputinformation to a subsequent input processor, wherein the process modelscomprise a powder spreading model, a powder melting model and a graingrowth model, and the mechanical response models comprise aself-consistent clustering analysis (SCA) model, a finite element method(FEM) model and/or a fatigue prediction model, wherein the powderspreading model and the powder melting model operably exchange geometryinformation of a powder bed using a stereolithography (STL)-formatsurface mesh representation; wherein the powder melting model operablypredicts a temperature profile in each volume of interest (VOI) with apowder melting simulation, wherein the temperature profile is written tothe database in a flat file format, and wherein a void space is set toan ambient temperature to distinguish said void space from a densematerial; wherein the grain growth model operably accesses thetemperature profile and void information from the database, and predictsgrain information based on the temperature profile and void information,wherein the grain information is written to the database along with thevoids, and wherein the dense material and the voids are identified basedon the temperature profile, and the voids and free surfaces are therebyincluded in the grain growth model; wherein the SCA and/or FEM modelsare used, based on the grain and void information provided at each pointvia the database, to predict stress-strain responses, wherein thestress-strain responses are written to the database; and wherein thefatigue prediction model operably accesses the stress-strain responsesat each point via the database, and computes a non-local potencyestimate based on the stress-strain responses, wherein the fatigue lifeof the highest potency location is computed to provide a scalar materialresponse metric.
 2. The framework of claim 1, wherein in operation, thepowder spreading model generates a particle packing configuration withinone powder layer with a particle spreading simulation, outputs theparticle packing configuration as a first STL file, and feeds theparticle packing configuration into the powder melting model, therebyreproducing powder spreading and melting; and the powder melting modelpredicts a solidified shape with the powder melting simulation, outputsthe solidified shape as a second STL file, and feeds the solidifiedshape into the powder spreading model to apply a new powder layer,thereby reproducing powder spreading.
 3. The framework of claim 2,wherein the manufacturing process of multiple layers and trackscomprises repeating said two simulations and the data exchange betweenthe powder spreading model and the powder melting model.
 4. Theframework of claim 1, wherein for the mechanical response models, asub-domain of particular interest of the full VOI for grain growth isselected as an input, the grain and void information provided at eachpoint via the database is assigned as the center point of each voxel ina regular, uniform, cuboid mesh, and said mesh is used for themechanical response models.
 5. The framework of claim 1, wherein thegrain information comprises phase and orientation.
 6. The framework ofclaim 1, wherein in operation, the powder melting model feeds thetemperature profiles and void information to the grain growth model viathe database, and the grain growth model feeds the grain and voidinformation to the SCA model via the database.
 7. The framework of claim1, wherein the powder spreading model is a discrete element method (DEM)powder spreading model.
 8. The framework of claim 1, wherein the powdermelting model is a computational fluid dynamics (CFD) powder meltingmodel.
 9. The framework of claim 1, wherein the grain growth model is acellular automaton (CA) grain growth model.
 10. The framework of claim9, wherein a material region is discretized by a set of cubic cells, andwherein the temperature history of each cell is determined from athermal-CFD simulation using a linear interpolation.
 11. The frameworkof claim 1, wherein the SCA model is an SCA crystal plasticity model.12. A method of integrated process-structure-property modeling fordesign optimization and/or performance prediction of a material system,comprising: exchanging geometry information of a powder bed between apowder spreading model and a powder melting model using astereolithography (STL)-format surface mesh representation; predicting atemperature profile in each volume of interest (VOI) by the powdermelting model with a powder melting simulation; writing the temperatureprofile to the database in a flat file format; and setting a void spaceto an ambient temperature to distinguish said void space from a densematerial; accessing the temperature profile and void information fromthe database and predicting grain information by a grain growth modelbased on the temperature profile and void information; writing the graininformation to the database along with the voids; identifying the densematerial and the voids on the temperature profile, thereby including thevoids and free surfaces in the grain growth model; predictingstress-strain responses by the SCA and/or FEM models based on the grainand void information provided at each point via the database; andwriting the stress-strain responses to the database; and accessing thestress-strain responses at each point via the database and computing anon-local potency estimate by a fatigue prediction model based on thestress-strain responses, wherein the fatigue life of the highest potencylocation is computed to provide a scalar material response metric. 13.The method of claim 12, wherein said exchanging the geometry informationcomprises: by the powder spreading model, generating a particle packingconfiguration within one powder layer with a particle spreadingsimulation, outputing the particle packing configuration as a first STLfile, and feeding the particle packing configuration into the powdermelting model, thereby reproducing powder spreading and melting; and bythe powder melting model, predicting a solidified shape with a powdermelting simulation, outputing the solidified shape as a second STL file,and feeding the solidified shape into the powder spreading model toapply a new powder layer, thereby reproducing powder spreading.
 14. Themethod of claim 13, wherein said exchanging the geometry informationfurther comprises repeating said two simulations and the data exchangebetween the powder spreading model and the powder melting model, forforming multiple layers and tracks.
 15. The method of claim 12, whereinthe particle packing configuration is generated using a discrete elementmethod (DEM).
 16. The method of claim 12, wherein the powder meltingmodel is a computational fluid dynamics (CFD) powder melting model. 17.The method of claim 12, wherein the grain growth model is a cellularautomaton (CA) grain growth model.
 18. The method of claim 12, whereinthe SCA model is an SCA crystal plasticity model.
 19. A non-transitorytangible computer-readable medium storing instructions which, whenexecuted by one or more processors, cause a system to perform a methodof integrated process-structure-property modeling for designoptimization and/or performance prediction of a material system, whereinthe method is in accordance with claim
 12. 20. A computational systemfor design optimization and/or performance prediction of a materialsystem, comprising: one or more computing devices comprising one or moreprocessors; and a non-transitory tangible computer-readable mediumstoring instructions which, when executed by the one or more processors,cause the one or more computing devices to perform a method ofintegrated process-structure-property modeling for design optimizationand/or performance prediction of a material system, wherein the methodis in accordance with claim
 12. 21. An integratedprocess-structure-property modeling framework for design optimizationand/or performance prediction of a material system, comprising a powderspreading model using a discrete element method (DEM) to generate apowder bed; a thermal-fluid flow model of the powder melting process topredict voids and temperature profile; a cellular automaton (CA) modelto simulate grain growth based on the temperature profile; and areduced-order micromechanics model to predict mechanical properties andfatigue resistance of resultant structures by resolving the voids andgrains.
 22. The framework of claim 19, wherein in the CA, a materialregion is discretized by a set of cubic cells, and wherein thetemperature history of each cell is determined from a thermal-CFD(computational fluid dynamics) simulation using a linear interpolation.23. The framework of claim 19, wherein each model incorporates basicmaterial information and data provided directly from previous models inthe framework to predict the mechanical response and estimated fatiguelife of critical microstructures given machine-relevant processingconditions.
 24. A method for implementation of multiresolution continuumtheory (MCT), comprising: decomposing deformation across differentmicrostructural length scales, by an MCT model; wherein the macroscaledeformation is described by conventional degrees of freedom and extradegrees of freedom are introduced to describe the deformation at amicroscale; employing a modified homogenization-based Gurson (GTN) modelat the macroscale to result in equations for a void volume fraction, aneffective plastic strain rate, and a matrix flow stress, wherein anadditive decomposition of a rate of deformation tensor into elastic andplastic parts allows for description of an objective rate of Cauchystress; solving the resulting equations at each scale using an iterativeNewton-Raphson scheme to update the void volume fraction, the effectiveplastic strain rate, and the matrix flow stress based on a newdeformation tensor, wherein the homogenized void growth behavior istaken to represent the response of the primary population of voids,including those initiated at inclusions, as the material is deformed;and constructing a plastic potential using microstress and microstresscouple at the microscale, wherein a deviatoric strain rate is defined interms of a relative deformation between the microscale and themacroscale, and solving equations of the plastic potential to obtainstress-strain profiles.
 25. The method of claim 24, wherein the materialstructure and the deformation field are resolved at each scale; theresulting internal power is a multi-field expression with contributionsfrom the average deformation at each scale; and the deformation behaviorat each scale is found by testing the micromechanical response.
 26. Themethod of claim 24, wherein the MCT model is implemented with a userelement subroutine including a FEA solver; user materials subroutinesfor micro and macro domains; an auxiliary function subroutine containingfunctions used by the user materials subroutines needed for the userelement subroutine, wherein the auxiliary function subroutine iscallable by the user material subroutines to finish the update of thestress and strain; a force subroutine for calculating the internal forceby calling the user materials subroutines and the auxiliary functionsubroutine; a parameter subroutine providing input parameters for theuser element subroutine; and a mass subroutine for computing a massmatrix for the user element subroutine, wherein the force subroutine,parameter subroutine and the mass subroutines are called by the userelement subroutine for solving the equations at each scale.
 27. Themethod of claim 26, wherein the auxiliary function subroutine comprisesGauss integration, computation of inertia and subroutines to calculatethe strain and strain rate.
 28. The method of claim 26, wherein theinput parameters include number of micro scales, number of Gauss pointsand number of history variables.
 29. The method of claim 24, beingapplicable to prediction of formation and propagation of a shear band ina material system.
 30. The method of claim 24, being applicable tosimulation of high speed metal cutting (HSMC) in a material system. 31.A non-transitory tangible computer-readable medium storing instructionswhich, when executed by one or more processors, cause a system toperform a method for implementation of the MCT, wherein the method is inaccordance with claim
 24. 32. A computational system for designoptimization and/or performance prediction of a material system,comprising: one or more computing devices comprising one or moreprocessors; and a non-transitory tangible computer-readable mediumstoring instructions which, when executed by the one or more processors,cause the one or more computing devices to perform a method forimplementation of the MCT, wherein the method is in accordance withclaim
 24. 33. A method of data-driven mechanistic modeling of a systemfor predicting high cycle fatigue (HCF) crack incubation life,comprising: generating a representation of the system with matrix,inclusion and void, wherein the representation comprises microstructurevolume elements (MVE) that are of building blocks of the system;obtaining strain contours in the matrix from a linear elastic analysisover a predefined set of boundary conditions; generating clusters from astrain solution by assigning voxels with similar strain concentrationtensor within the representation of the material system to one of theclusters with a unique ID, using k-means clustering; computing a plasticshear strain field of the material system by solving theLippmann-Schwinger equation using the generated clusters with a crystalplasticity (CP) model; and predicting the fatigue crack incubation lifeof the system using a fatigue indicating parameter (FIP) based on theplastic shear strain field.
 34. A non-transitory tangiblecomputer-readable medium storing instructions which, when executed byone or more processors, cause a system to perform a data-drivenmechanistic modeling method for predicting high cycle fatigue (HCF)crack incubation life, wherein the method is in accordance with claim33.
 35. A computational system for design optimization and/orperformance prediction of a material system, comprising: one or morecomputing devices comprising one or more processors; and anon-transitory tangible computer-readable medium storing instructionswhich, when executed by the one or more processors, cause the one ormore computing devices to perform a data-driven mechanistic modelingmethod for predicting high cycle fatigue (HCF) crack incubation life,wherein the method is in accordance with claim 33.